Abstract

Retrieving the inherent optical properties of water from remote sensing multispectral reflectance measurements is difficult due to both the complex nature of the forward modeling and the inherent nonlinearity of the inverse problem. In such cases, neural network (NN) techniques have a long history in inverting complex nonlinear systems. The process we adopt utilizes two NNs in parallel. The first NN is used to relate the remote sensing reflectance at available MODIS-visible wavelengths (except the 678nm fluorescence channel) to the absorption and backscatter coefficients at 442nm (peak of chlorophyll absorption). The second NN separates algal and nonalgal absorption components, outputting the ratio of algal-to-nonalgal absorption. The resulting synthetically trained algorithm is tested using both the NASA Bio-Optical Marine Algorithm Data Set (NOMAD), as well as our own field datasets from the Chesapeake Bay and Long Island Sound, New York. Very good agreement is obtained, with R2 values of 93.75%, 90.67%, and 86.43% for the total, algal, and nonalgal absorption, respectively, for the NOMAD. For our field data, which cover absorbing waters up to about 6m1, R2 is 91.87% for the total measured absorption.

© 2011 Optical Society of America

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2008 (1)

2007 (2)

A. Gitelson, J. F. Schalles, and C. M. Hladik, “Remote chlorophyll-a retrieval in turbid, productive estuaries: Chesapeake Bay case study,” Remote Sens. Environ. 109, 464–472 (2007).
[CrossRef]

A. Morel, B. Gentili, H. Claustre, M. Babin, A. Bricaud, J. Ras, and F. Tièche, “Optical properties of the “clearest” natural waters,” Limnol. Oceanogr. 52, 217–229 (2007).
[CrossRef]

2005 (2)

P. J. Werdell and S. W. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122–140(2005).
[CrossRef]

P. Wang, E. S. Boss, and C. Roesler, “Uncertainties of inherent optical properties obtained from semianalytical inversions of ocean color,” Appl. Opt. 44, 4074–4085 (2005).
[CrossRef] [PubMed]

2004 (3)

Z. P. Lee and K. L. Carder, “Absorption spectrum of phytoplankton pigments derived from hyperspectral remote-sensing reflectance,” Remote Sens. Environ. 89, 361–368 (2004).
[CrossRef]

F. Aires, C. Prigent, and W. B. Rossow, “Neural network uncertainty assessment using Bayesian statistics: a remote sensing application,” Neural Comput. 16, 2415–2458 (2004).
[CrossRef] [PubMed]

A. Tanaka, M. Kishino, R. Doerffer, H. Schiller, T. Oishi, and T. Kubota, “Development of a neural network algorithm for retrieving concentrations of chlorophyll, suspended matter and yellow substance from radiance data of the ocean color and temperature scanner,” J. Oceanogr. 60, 519–530 (2004).
[CrossRef]

2003 (3)

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and oceanic waters as related to the particle mass concentration,” Limnol. Oceanogr. 48, 843–859 (2003).
[CrossRef]

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211–3231 (2003).
[CrossRef]

A. Albert and C. D. Mobley, “An analytical model for subsurface irradiance and remote sensing reflectance in deep and shallow case-2 waters,” Opt. Express 11, 2873–2890(2003).
[CrossRef] [PubMed]

2002 (4)

S. B. Hooker, G. Lazin, G. Zibordi, and S. McLean, “An evaluation of above- and in-water methods for determining water-leaving radiances,” J. Atmos. Ocean. Technol. 19, 486–515(2002).
[CrossRef]

A. M. Ciotti, M. R. Lewis, and J. J. Cullen, “Assessment of the relationships between dominant cell size in natural phytoplankton communities and the spectral shape of the absorption coefficient,” Limnol. Oceanogr. 47, 404–417 (2002).
[CrossRef]

T. Oishi, Y. Takahashi, A. Tanaka, M. Kishino, and A. Tsuchiya, “Relation between the backward as well as total scattering coefficients and the volume scattering functions by cultured phytoplankton,” J. School Mar. Sci. Technol. Tokai Univ. 53, 1–15 (2002).

Z. P. Lee, K. L. Carder, and R. A. Armone, “Deriving inherent optical properties from water color: a multiband quasi-analytical algorithm for optically deep waters,” Appl. Opt. 41, 5755–5772 (2002).
[CrossRef] [PubMed]

2001 (4)

F. Aires, C. Prigent, W. B. Rossow, and M. Rothstein, “A neural network approach including first guess for retrieval of atmospheric water vapor, cloud liquid water path, surface temperature, and emissivities over land from satellite microwave observations,” J. Geophys. Res. 106, 14887–14907 (2001).
[CrossRef]

H. Loisel, D. Stramski, B. G. Mitchell, F. Fell, V. Fournier-Sicre, B. Lemasle, and M. Babin, “Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling,” Appl. Opt. 40, 2384–2397 (2001).
[CrossRef]

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. 40, 2929–2945(2001).
[CrossRef]

A. Morel and S. Maritorena, “Bio-optical properties of oceanic waters: a reappraisal,” J. Geophys. Res. 106, 7163–7180(2001).
[CrossRef]

1999 (4)

Z. P. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters: 2. deriving bottom depths and water properties by optimization,” Appl. Opt. 38, 3831–3843 (1999).
[CrossRef]

H. Schiller and R. Doerffer, “Neural network for emulation of an inverse mode-operational derivation of case II water properties from MERIS data,” Int. J. Remote Sens. 20, 1735–1746(1999).
[CrossRef]

L. Gross, S. Thiria, and R. Frouin, “Applying artificial neural network methodology to ocean color remote sensing,” Eco. Model. 120, 237–246 (1999).
[CrossRef]

K. L. Carder, F. R. Chen, Z. P. Lee, S. K. Hawes, and D. Kamykowski, “Semianalytic moderate-resolution imaging spectrometer algorithms for chlorophyll-a and absorption with bio-optical domains based on nitrate-depletion temperatures,” J. Geophys. Res. 104, 5403–5421 (1999).
[CrossRef]

1998 (3)

M. Sydor, R. Arnone, R. W. Gould, G. E. Terrie, S. D. Ladner, and C. G. Wood, “Remote-sensing technique for determination of the volume absorption coefficient of turbid water,” Appl. Opt. 37, 4944–4950 (1998).
[CrossRef]

J. O’Reilly, S. Maritorena, B. G. Mitchell, D. Siegel, K. L. Carder, S. Garver, M. Kahru, and C. McClain, “Ocean color chlorophyll algorithms for SeaWiFS,” J. Geophys. Res. 103, 24937–24953 (1998).
[CrossRef]

Z. P. Lee, K. L. Carder, R. G. Steward, T. G. Peacock, C. O. Davis, and J. S. Patch, “An empirical algorithm for light absorption by ocean water based on color,” J. Geophys. Res. 103, 27967–27978 (1998).
[CrossRef]

1997 (2)

A. H. Garver and D. A. Siegel, “Inherent optical property inversion of ocean color spectra and its biogeochemical interpretation. 1. time series from the Sargasso Sea,” J. Geophys. Res. 102, 18607–18625 (1997).
[CrossRef]

R. Pope and E. Fry, “Absorption spectrum 380–700 nm of pure waters: II. integrating cavity measurements,” Appl. Opt. 36, 8710–8723 (1997).
[CrossRef]

1996 (1)

F. E. Hoge and P. E. Lyon, “Satellite retrieval of inherent optical properties by linear matrix inversion of oceanic radiance models: an analysis of model and radiance measurement errors,” J. Geophys. Res. 101, 16631–16648 (1996).
[CrossRef]

1995 (3)

C. S. Roesler and M. J. Perry, “In situ phytoplankton absorption, fluorescence emission, and particulate backscattering spectra determined from reflectance,” J. Geophys. Res. 100, 13279–13294 (1995).
[CrossRef]

J. W. Campbell, “The lognormal distribution as a model for bio-optical variability in the sea,” J. Geophys. Res. 100, 13237–13254 (1995).
[CrossRef]

A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13321–13332 (1995).
[CrossRef]

1994 (1)

1992 (1)

D. J. C. MacKay, “Bayesian interpolation,” Neural Comput. 4, 415–447 (1992).
[CrossRef]

1991 (1)

A. Morel, “Light and marine photosynthesis: a spectral model with geochemical and climatological implications,” Progr. Oceanogr. 26, 263–306 (1991).
[CrossRef]

1989 (2)

K. Horrnik, M. Stinchcombe, and H. White, “multilayer feedforward networks are universal approximators,” Neural Netw. 2, 359–366 (1989).
[CrossRef]

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Syst. 2, 303–314 (1989).
[CrossRef]

1988 (2)

A. Morel, “Optical modeling of the upper ocean in relation to its biogenous matter content (case I waters),” J. Geophys. Res. 93, 10749–10768 (1988).
[CrossRef]

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

1983 (1)

1977 (1)

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr. 22, 709–722 (1977).
[CrossRef]

1963 (1)

D. W. Marquardt, “An algorithm for the least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

1944 (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168(1944).

Ahmed, S.

Aires, F.

F. Aires, C. Prigent, and W. B. Rossow, “Neural network uncertainty assessment using Bayesian statistics: a remote sensing application,” Neural Comput. 16, 2415–2458 (2004).
[CrossRef] [PubMed]

F. Aires, C. Prigent, W. B. Rossow, and M. Rothstein, “A neural network approach including first guess for retrieval of atmospheric water vapor, cloud liquid water path, surface temperature, and emissivities over land from satellite microwave observations,” J. Geophys. Res. 106, 14887–14907 (2001).
[CrossRef]

Albert, A.

Armone, R. A.

Arnone, R.

Austin, R. W.

R. W. Austin and T. J. Petzold, “The determination of the diffuse attenuation coefficient of sea water using the coastal zone color scanner,” in Oceanography from Space, J.F. R.Gower, ed. (Plenum, 1981).
[CrossRef]

Babin, M.

A. Morel, B. Gentili, H. Claustre, M. Babin, A. Bricaud, J. Ras, and F. Tièche, “Optical properties of the “clearest” natural waters,” Limnol. Oceanogr. 52, 217–229 (2007).
[CrossRef]

M. Babin, A. Morel, V. Fournier-Sicre, F. Fell, and D. Stramski, “Light scattering properties of marine particles in coastal and oceanic waters as related to the particle mass concentration,” Limnol. Oceanogr. 48, 843–859 (2003).
[CrossRef]

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211–3231 (2003).
[CrossRef]

H. Loisel, D. Stramski, B. G. Mitchell, F. Fell, V. Fournier-Sicre, B. Lemasle, and M. Babin, “Comparison of the ocean inherent optical properties obtained from measurements and inverse modeling,” Appl. Opt. 40, 2384–2397 (2001).
[CrossRef]

A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13321–13332 (1995).
[CrossRef]

Bailey, S. W.

P. J. Werdell and S. W. Bailey, “An improved in-situ bio-optical data set for ocean color algorithm development and satellite data product validation,” Remote Sens. Environ. 98, 122–140(2005).
[CrossRef]

Baker, K. S.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

Boss, E. S.

Bricaud, A.

A. Morel, B. Gentili, H. Claustre, M. Babin, A. Bricaud, J. Ras, and F. Tièche, “Optical properties of the “clearest” natural waters,” Limnol. Oceanogr. 52, 217–229 (2007).
[CrossRef]

M. Babin, D. Stramski, G. M. Ferrari, H. Claustre, A. Bricaud, G. Obolensky, and N. Hoepffner, “Variations in the light absorption coefficients of phytoplankton, nonalgal particles, and dissolved organic matter in coastal waters around Europe,” J. Geophys. Res. 108, 3211–3231 (2003).
[CrossRef]

D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. 40, 2929–2945(2001).
[CrossRef]

A. Bricaud, M. Babin, A. Morel, and H. Claustre, “Variability in the chlorophyll-specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. 100, 13321–13332 (1995).
[CrossRef]

Broenkow, W. W.

Brown, J. W.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36(1983).
[CrossRef] [PubMed]

Brown, O. B.

H. R. Gordon, O. B. Brown, R. H. Evans, J. W. Brown, R. C. Smith, K. S. Baker, and D. K. Clark, “A semi-analytic radiance model of ocean color,” J. Geophys. Res. 93, 10909–10924(1988).
[CrossRef]

H. R. Gordon, D. K. Clark, J. W. Brown, O. B. Brown, R. H. Evans, and W. W. Broenkow, “Phytoplankton pigment concentrations in the Middle Atlantic Bight: comparison of ship determinations and CZCS estimates,” Appl. Opt. 22, 20–36(1983).
[CrossRef] [PubMed]

Bukata, R. P.

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Figures (19)

Fig. 1
Fig. 1

Normalized distributions of the simulated dataset parameters (black curve) and the NOMAD (gray curve). Noticeably, the measurements fall within the range of our simulations.

Fig. 2
Fig. 2

Relationships of the three components at the reference wavelength used in the simulations according to the distribution functions used in our dataset. The black dots indicate the simulated datasets, while the gray points indicate the measurements of the NOMAD. We can clearly observe that the NOMAD measurements fall within our simulations.

Fig. 3
Fig. 3

Comparison between in situ and simulated dataset as indicated in IOCCG report 5. Except for very clear waters where the R rs ( 488 ) / R rs ( 550 ) is greater than about 5, all the other ranges are within our simulations.

Fig. 4
Fig. 4

Performance of the NN on the part of our simulated dataset that was not used in the training stage. Inverted a ( 442 ) m 1 (left) and b b ( 442 ) m 1 (right) (x axis) plotted against the “known” values from our simulated dataset. Noise levels, ε, up to 20% (the top row is ε = 0 % ) were added at each R rs . The corresponding statistics are shown in Tables 1, 2, 3.

Fig. 5
Fig. 5

Performance of the NN on the part of our simulated dataset that was not used in the training stage. Inverted a ph ( 442 ) m 1 (left) and a dg ( 442 ) m 1 (right) (x axis) plotted against the “known” values for these parameters from our simulated dataset. Noise levels, ε, up to 20% (the top row is ε = 0 % ) were added at each R rs . The gray dots indicate the cases where a ph ( 442 ) is less than 10% of a dg ( 442 ) . The corresponding statistics are shown in Tables 1, 2, 3 below.

Fig. 6
Fig. 6

Inverted IOP from the NN (x axis) plotted against the in situ values from the NOMAD for a ( 442 ) m 1 , b b ( 442 ) m 1 , and K d ( 442 ) m 1 . The lower figures illustrate the relationship of a ( 442 ) m 1 and b b ( 442 ) m 1 (left) and K d { a ( 442 ) , b b ( 442 ) } m 1 from Eq. (30), right. The statistics are shown in Table 4.

Fig. 7
Fig. 7

Inverted IOPs from the NN (x axis) plotted against the in situ values for our field dataset for a ( 442 ) m 1 (top) and b b ( 442 ) m 1 (bottom). The statistics are shown in the parenthesis in Table 4.

Fig. 8
Fig. 8

Inverted IOPs from the NN (x axis) plotted against the in situ values from the NOMAD for a ph ( 442 ) m 1 and a dg ( 442 ) m 1 . The first row represents the outputs of the NN before “fitting” the output with the measurements (see text). The second row represents the results after adjustment with the field data. Statistics are shown in Table 4.

Fig. 9
Fig. 9

Retrievals of the SAA, QAA, and NN (x axis) versus in situ measurement for a ( 442 ) m 1 . Statistics are shown in Table 5.

Fig. 10
Fig. 10

Retrievals of the SAA, QAA, and NN (x axis) versus in situ measurement for a ph ( 442 ) m 1 . Statistics are shown in Table 6.

Fig. 11
Fig. 11

Retrievals of the SAA, QAA, and NN (x axis) versus in situ measurement for a dg ( 442 ) m 1 . Statistics are shown in Table 7.

Fig. 12
Fig. 12

Retrievals of the SAA, QAA, and NN (x axis) versus in situ measurement for b b ( 442 ) m 1 . Statistics are shown in Table 8.

Fig. 13
Fig. 13

Retrievals of the NN of the total absorption coefficient, a ( 442 ) m 1 (top), and the total backscattering coefficient b b ( 442 ) m 1 (bottom). The color bar is adjusted in the smaller images on the left to make the turbid regions visible.

Fig. 14
Fig. 14

NN retrieval of a ( 442 ) m 1 x axis plotted against the SAA (top) and QAA (bottom). R 2 = 99.16 with a slope of 0.9421 and an intercept of 0.0945 for the SAA, and R 2 = 0.9915 , a slope of 1.0365 and an intercept of 0.0359 for the QAA. All the available retrievals for all algorithms are shown.

Fig. 15
Fig. 15

Retrievals of the NN of the algal absorption coefficient, a ph ( 442 ) m 1 (top), and the nonalgal absorption coefficient a dg ( 442 ) m 1 (bottom). The color bar is adjusted in the smaller images on the left to make the turbid regions visible.

Fig. 16
Fig. 16

The NN retrieval of a ph ( 442 ) m 1 x axis plotted against the SAA (top) and QAA (bottom). R 2 = 0.6893 with a slope of 1.0641 and an intercept of 0.2264 for the SAA, and R 2 = 0.5949 , a slope of 1.0802 and an intercept of 0.2648 for the QAA. All the available retrievals for all algorithms are shown.

Fig. 17
Fig. 17

NN retrieval of a dg ( 442 ) m 1 x axis plotted against the SAA (top) and QAA (bottom). R 2 = 0.9806 with a slope of 0.8569 and an intercept of 0.1179 for the SAA, and R 2 = 0.9620 , a slope of 1.0143 and an intercept of 0.0948 for the QAA. All the available retrievals for all algorithms are shown.

Fig. 18
Fig. 18

NN retrieval of b b ( 442 ) m 1 x axis plotted against the SAA left and QAA right. R 2 = 0.9822 with a slope of 0.8854 and an intercept of 0.2982 for the SAA, and R 2 = 0.9866 , a slope of 0.9426 and an intercept of 0.18 for the QAA. All the available retrievals for all algorithms are shown.

Fig. 19
Fig. 19

MODIS [Chl] mg / m 3 product x axis plotted against the NN retrieval of a ph ( 442 ) m 1 divided by the MODIS [Chl] (specific absorption of algae). Specific absorptions of less than 0.01 m 1 , which consistently appear in the higher [Chl] are probably due to CDOM-contaminated [Chl] retrievals. Three models are shown from [31] (cyan—middle curve), [32] (red—lower curve), [57] (magenta—top curve).

Tables (11)

Tables Icon

Table 1 Statistics of Comparison for Figs. 4, 5 without Noise a

Tables Icon

Table 2 Statistics of Comparison for Figs. 4, 5 When 10% Noise Is Added at Each R rs a

Tables Icon

Table 3 Statistics of Comparison for Figs. 4, 5 When 20% Noise Is Added at Each R rs a

Tables Icon

Table 4 Statistics of Comparison for Figs. 6, 7, 8 a

Tables Icon

Table 5 Statistics of Comparison of the Three Algorithms for a ( 442 ) m 1 a

Tables Icon

Table 6 Statistics of Comparison of the Three Algorithms for a ph ( 442 ) m 1 a

Tables Icon

Table 7 Statistics of Comparison of the Three Algorithms for a dg ( 442 ) m 1 a

Tables Icon

Table 8 Statistics of Comparison of the Three Algorithms for b b ( 442 ) m 1 a

Tables Icon

Table 9 Mean and Standard Deviation of the Inputs for the a pg ( 442 ) m 1 and b bp ( 442 ) m 1 Network

Tables Icon

Table 10 Mean and Standard Deviation of the Inputs for the R a dg a ph ( 442 ) Network

Tables Icon

Table 11 Mean and Standard Deviation of the Outputs in the Simulated Dataset

Equations (38)

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

y m = y ( x ) + ε .
D ( x ^ , x ) P ( x , ε ) ,
R rs ( λ ) = g [ a ( λ ) , β ( λ ) , θ Sun , θ sensor , ϕ azimuth ] sr 1 ,
a ( λ ) = a w ( λ ) + a ph ( λ ) + a dm ( λ ) + a g ( λ ) m 1 ,
b b ( λ ) = b b w ( λ ) + b b ph ( λ ) + b b dm ( λ ) m 1 ,
a ph * ( λ ) = S f × a pico * ( λ ) + ( 1 S f ) × a micro * ( λ ) m 2 mg 1 ,
a ph ( λ ) = a ph * ( λ ) × [ Chl ] P m 1 ,
b ph ( λ ) = c ph ( λ ) a ph ( λ ) m 1 .
c ph ( λ ) = c ph ( 550 ) × ( 550 λ ) Y ph m 1 ,
c ph ( 550 ) = ρ × [ Chl ] P m 1
b b ph ( λ ) = b ˜ ph b ph ( λ ) m 1 ,
a dm ( λ ) = a dm ( 412 ) × exp [ S dm × ( 412 λ ) ] m 1 ,
a dm ( 412 ) = a dm * ( 412 ) × [ NAP ] m 1 .
b dm ( λ ) = b dm ( 550 ) ( 550 λ ) Y dm m 1 ,
b b dm ( λ ) = b ˜ dm b dm ( λ ) m 1 ,
b dm ( 550 ) = b dm * ( 550 ) × [ NAP ] m 1 ,
a g ( λ ) = a g ( 412 ) × exp [ S g × ( 412 λ ) ] m 1 ,
[ Chl ] = 0.02 + 70 × exp ( χ 1.3 ) × Ψ 1 mg m 3 ,
[ NAP ] = 0.02 + 50 × exp ( χ 1.4 ) × Ψ 2 g m 3 ,
a g ( 412 ) = 0.001 + 6 × exp ( χ 1.2 ) × Ψ 3 m 1 ,
a pg ( 442 ) = a dg ( 442 ) + a ph ( 442 ) m 1 .
a ph ( 442 ) = a pg ( 442 ) 1 + 1 R a dg a ph ( 442 ) m 1 .
a dg ( 442 ) = a pg ( 442 ) a ph ( 442 ) m 1 .
input ( i ) = log 10 ( R rs ( i ) ) μ input ( i ) σ input ( i ) ,
output ( 1 ) = log 10 ( a pg ) μ o ( 1 ) σ o ( 1 ) ,
output ( 2 ) = log 10 ( b bp ) μ o ( 2 ) σ o ( 2 ) ,
output = log 10 ( a ph a dg ) μ o ( 3 ) σ o ( 3 ) .
RMSE log 10 = { 1 N i = 1 N [ log 10 ( x ^ i ) log 10 ( x i ) ] 2 } 1 2 ,
e = 10 RMSE log 10 1.
K d { a ( 442 ) , b b ( 442 ) } = 1 . 2574 × { a ( 442 ) + b b ( 442 ) } 0 . 0030 m 1 .
R a dg a ph ( 442 ) = a ph ( 442 ) a pg ( 442 ) a ph ( 442 ) .
f ( n ) = 2 1 + e 2 n 1 ,
n = [ 0 . 0263 0 . 0974 0 . 0732 0 . 1207 0 . 0559 0 . 1136 0 . 2866 0 . 3382 0 . 0898 0 . 4422 0 . 0188 1 . 0006 0 . 7189 0 . 0737 0 . 7047 0 . 6362 0 . 1679 1 . 9526 0 . 1271 0 . 398 0 0 . 2976 0 . 3444 0 . 0955 1 . 1113 0 . 0797 0 . 1104 0 . 0283 0 . 2284 0 . 1133 0 . 7287 0 . 007 0 0 . 1419 0 . 0126 0 . 1071 0 . 0784 0 . 0616 ] × input + [ 0 . 0618 1 . 9081 1 . 6717 0 . 1662 1 . 9662 0 . 7745 ] ,
n = [ 0 . 0625 0 . 1401 0 . 1135 0 . 204 0 . 0172 0 . 6069 1 . 1289 0 . 2884 0 . 2429 0 . 1419 0 . 4505 0 . 0433 0 . 4030 0 . 0423 0 . 155 0 . 0097 0 . 1402 0 . 8283 0 . 6160 0 . 3984 0 . 4717 0 . 2444 0 . 4624 0 . 1867 0 . 2288 0 . 1289 0 . 6219 0 . 2947 0 . 2731 1 . 0645 0 . 2884 0 . 3029 0 . 4365 0 . 1793 0 . 2644 1 . 0189 ] × input + [ 0 . 9316 0 . 2837 0 . 3768 0 . 5398 1 . 0015 0 . 5281 ] ,
y 1 = [ 0 . 2411 0 . 1989 0 . 2459 0 . 4511 0 . 2136 2 . 4228 1 . 3594 0 . 6151 0 . 2335 0 . 4726 1 . 9273 0 . 0266 ] × f ( n ) + [ 1 . 3425 1 . 0 179 ] ,
y 2 = [ 0 . 5533 0 . 3642 0 . 6332 0 . 9063 0 . 8372 0 . 7898 ] × f ( n ) + [ 0 . 3642 ] ,
{ a pg ( 442 ) = 10 { σ O ( 1 ) × y 1 ( 1 ) μ O ( 1 ) } b bp ( 442 ) = 10 { σ O ( 2 ) × y 1 ( 2 ) μ O ( 2 ) } }
R a dg a ph ( 442 ) = 10 { σ O ( 3 ) × y 2 μ O ( 3 ) }

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