Abstract

Inherent optical property (IOP) spectral models for the phytoplankton absorption coefficient, chromophoric dissolved organic matter (CDOM) absorption coefficient, and total constituent backscattering (TCB) coefficient are linear in the reference wavelength IOP and nonlinear in the spectral parameters. For example, the CDOM absorption coefficient IOP a CDOMi) = a CDOMref)exp[-Si - λref)] is linear in a CDOMref) and nonlinear in S. Upon linearization by Taylor’s series expansion, it is shown that spectral model parameters, such as S, can be concurrently accommodated within the same conventional linear matrix formalism used to retrieve the reference wavelength IOP’s. Iteration is used to adjust for errors caused by truncation of the Taylor’s series expansion. Employing an iterative linear matrix inversion of a water-leaving radiance model, computer simulations using synthetic data suggest that (a) no instabilities or singularities are introduced by the linearization and subsequent matrix inversion procedures, (b) convergence to the correct value can be expected only if starting values for a model parameter are within certain specific ranges, (c) accurate retrievals of the CDOM slope S (or the phytoplankton Gaussian width g) are generally reached in 3–20 iterations, (d) iterative retrieval of the exponent n of the TCB wavelength ratio spectral model is not recommended because the starting values must be within ∼±5% of the correct value to achieve accurate convergence, and (e) concurrent retrieval of S and g (simultaneously with the phytoplankton, CDOM, and TCB coefficient IOP’s) can be accomplished in a 5 × 5 iterative matrix inversion if the starting values for S and g are carefully chosen to be slightly higher than the expected final retrieved values.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. E. Hoge, 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, 16,631–16,648 (1996).
    [CrossRef]
  2. C. R. Wylie, Advanced Engineering Mathematics, 3rd ed. (McGraw-Hill, New York, 1960), Chap. 4, pp. 136–137.
  3. J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.
  4. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.
  5. Z. P. Lee, K. L. Carder, S. K. Hawes, R. G. Steward, T. G. Peacock, C. O. Davis, “Model for the interpretation of hyperspectral remote-sensing reflectance,” Appl. Opt. 33, 5721–5732 (1994).
    [CrossRef] [PubMed]
  6. R. A. Maffione, D. R. Dana, J. M. Voss, “Spectral dependence of optical backscattering in the ocean,” presented at OSA Annual Meeting, Portland, Oregon, 10–15 September 1995, paper MDD4, p. 56.

1996 (1)

F. E. Hoge, 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, 16,631–16,648 (1996).
[CrossRef]

1994 (1)

Aiken, J.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

Carder, K. L.

Clark, D. K.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

Dana, D. R.

R. A. Maffione, D. R. Dana, J. M. Voss, “Spectral dependence of optical backscattering in the ocean,” presented at OSA Annual Meeting, Portland, Oregon, 10–15 September 1995, paper MDD4, p. 56.

Davis, C. O.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.

Hawes, S. K.

Hoge, F. E.

F. E. Hoge, 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, 16,631–16,648 (1996).
[CrossRef]

Hooker, S. B.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

Lee, Z. P.

Lyon, P. E.

F. E. Hoge, 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, 16,631–16,648 (1996).
[CrossRef]

Maffione, R. A.

R. A. Maffione, D. R. Dana, J. M. Voss, “Spectral dependence of optical backscattering in the ocean,” presented at OSA Annual Meeting, Portland, Oregon, 10–15 September 1995, paper MDD4, p. 56.

Moore, G. F.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

Peacock, T. G.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.

Steward, R. G.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.

Trees, C. C.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.

Voss, J. M.

R. A. Maffione, D. R. Dana, J. M. Voss, “Spectral dependence of optical backscattering in the ocean,” presented at OSA Annual Meeting, Portland, Oregon, 10–15 September 1995, paper MDD4, p. 56.

Wylie, C. R.

C. R. Wylie, Advanced Engineering Mathematics, 3rd ed. (McGraw-Hill, New York, 1960), Chap. 4, pp. 136–137.

Appl. Opt. (1)

J. Geophys. Res. (1)

F. E. Hoge, 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, 16,631–16,648 (1996).
[CrossRef]

Other (4)

C. R. Wylie, Advanced Engineering Mathematics, 3rd ed. (McGraw-Hill, New York, 1960), Chap. 4, pp. 136–137.

J. Aiken, G. F. Moore, C. C. Trees, S. B. Hooker, D. K. Clark, “The SeaWiFS CZCS-type pigment algorithm,” , S. B. Hooker, E. R. Firestone, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 1995), Vol. 29, p. 1.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in Fortran, 2nd ed. (Cambridge U. Press, New York, 1992), Chap. 15, p. 672.

R. A. Maffione, D. R. Dana, J. M. Voss, “Spectral dependence of optical backscattering in the ocean,” presented at OSA Annual Meeting, Portland, Oregon, 10–15 September 1995, paper MDD4, p. 56.

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

For the exponent n of the TCB coefficient wavelength ratio spectral model: the percentage of the total data set (∼10,000 radiance spectra) that accurately converged when the initial or starting value n 0 was a given percentage Δ away from the actual n value used to generate the radiance spectrum. The starting value for n 0 must be within a few percent of the actual value of n to converge to the exact n value. These data suggest that linearization and iterative matrix inversion not be used for retrieval of the exponent n of the wavelength ratio backscatter model.

Fig. 2
Fig. 2

For the spectral slope S of the CDOM exponential spectral model: the percentage of the total data set (∼10,000 radiance spectra) that accurately converged when the initial or starting value S 0 was a given percentage Δ away from the actual S value used to generate the radiance spectrum. If the starting value for S 0 was >-25% from the actual value of S, then almost all iterative inversions successfully converged to within 1% of the S value used to generate the radiance spectrum. These data suggest that the linearization procedure is likely to retrieve a correct S when the starting value S 0 is chosen in the high range of expected S values.

Fig. 3
Fig. 3

For the spectral width g of the phytoplankton Gaussian spectral model: the percentage of the total data set (∼10,000 radiance spectra) that accurately converged when the initial or starting value g 0 was a given percentage Δ away from the actual g value. If the starting value for g 0 was within -55% and 55% of the actual value of g, then all iterative inversions successfully converged to within 1% of the g value used to generate the radiance spectrum. These data suggest that the linearization procedure is likely to retrieve a correct spectral width g when the starting value g 0 is chosen in the low range of expected values of g.

Fig. 4
Fig. 4

For width g of the phytoplankton Gaussian spectral model and slope S of the CDOM exponential spectral model: the percentage of the total data set (∼10,000 radiance spectra) that accurately converged when the initial or starting values g 0 and S 0 were a given percentage Δ away from the actual g and S values used to generate the radiance data. Correct convergence is found within the 95–100% contour region. These data suggest that the linearization procedure is more likely to retrieve both a correct spectral width g and a CDOM slope S when starting values g 0 and S 0 are chosen to be larger than their expected values.

Equations (6)

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

aphλgexp-λi-λg2/2g2+adλdexp-Sλi-λd+bbtλbλb/λinvλi=-awλi-bbwλivλi
fx, y=fx0, y0+f/xx0,y0x-x0+f/yx0,y0y-y0+1/22f/x2x0,y0x-x02+22f/xyx-x0y-y0+2f/y2x0,y0y-y02+.
bbtλbλb/λin=λb/λin0bbtλb+bbt0λbλb/λin0n-n0lnλb/λi+,
adλdexp-Sλi-λd=adλdexp-S0λi-λd-ad0λdexp-S0λi-λdλi-λdS-S0+,
aphλgexp-λi-λg2/2g2=aphλgexp-λi-λg02/2g02+aph0λgλi-λg2g0-3×g-g0exp-λi-λg02/2g02+.  
aphλgexp-λi-λg02/2g02+aph0λgλi-λg2g0-3×g-g0exp-λi-λg02/2g02+adλdexp-S0λi-λd-ad0λdexp-S0λi-λd×λi-λdS-S0+bbtλbλb/λin0vλi+n-n0bbt0λbλb/λin0 lnλb/λivλi=-awλi-bbwλivλi.

Metrics