Abstract

The use of the Mahalanobis distance in a lookup table approach to retrieval of in-water Inherent Optical Properties (IOPs) led to significant improvements in the accuracy of the retrieved IOPs, as high as 50% in some cases, with an average improvement of 20% over a wide range of case II waters. Previous studies have shown that inherent noise in hyperspectral data can cause significant errors in the retrieved IOPs. For LUT-based retrievals that rely on spectrum matching, the particular metric used for spectral comparisons has a significant effect on the accuracy of the results, especially in the presence of noise in the data. In this study, we have compared the Euclidean distance and the Mahalanobis distance as metrics for spectral comparison. In addition to providing justification for the preference of the Mahalanobis Distance over the Euclidean Distance, we have also included a statistical description of noisy hyperspectral data.

© 2013 OSA

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References

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    [CrossRef]
  2. R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC Press, 1995).
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    [CrossRef]
  4. A. Gitelson, “The peak near 700 nm on radiance spectra of algae and water - relationships of its magnitude and position with chlorophyll concentration,” Int. J. Remote Sens.13(17), 3367–3373 (1992).
    [CrossRef]
  5. G. Dall’Olmo and A. A. Gitelson, “Effect of bio-optical parameter variability on the remote estimation of chlorophyll-a concentration in turbid productive waters: experimental results,” Appl. Opt.44(3), 412–422 (2005).
    [CrossRef] [PubMed]
  6. Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
    [CrossRef] [PubMed]
  7. C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
    [CrossRef]
  8. W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India2(1), 49–55 (1936).
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    [CrossRef]
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2012 (1)

2011 (2)

R. L. Lucke, M. Corson, N. R. McGlothlin, S. D. Butcher, D. L. Wood, D. R. Korwan, R. R. Li, W. A. Snyder, C. O. Davis, and D. T. Chen, “Hyperspectral Imager for the Coastal Ocean: instrument description and first images,” Appl. Opt.50(11), 1501–1516 (2011).
[CrossRef] [PubMed]

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

2010 (1)

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

2009 (2)

J. Hedley, C. Roelfsema, and S. Phinn, “Efficient radiative transfer model inversion for remote sensing applications,” Remote Sens. Environ.113(11), 2527–2532 (2009).
[CrossRef]

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

2005 (2)

2000 (1)

1992 (1)

A. Gitelson, “The peak near 700 nm on radiance spectra of algae and water - relationships of its magnitude and position with chlorophyll concentration,” Int. J. Remote Sens.13(17), 3367–3373 (1992).
[CrossRef]

1989 (2)

C. D. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind roughened surfaces,” Limnol. Oceanogr.34(8), 1473–1483 (1989).
[CrossRef]

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

1977 (1)

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr.22(4), 709–722 (1977).
[CrossRef]

1936 (1)

P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India2(1), 49–55 (1936).

Ahmad, Z.

Bissett, W. P.

Bowles, J. H.

Butcher, S. D.

Carder, K. L.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

Chen, D. T.

Chen, J.

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

Corson, M.

Corson, M. R.

Dall’Olmo, G.

Davis, C. O.

Downes, T. V.

Fukushima, T.

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

Gao, B. C.

Gitelson, A.

A. Gitelson, “The peak near 700 nm on radiance spectra of algae and water - relationships of its magnitude and position with chlorophyll concentration,” Int. J. Remote Sens.13(17), 3367–3373 (1992).
[CrossRef]

Gitelson, A. A.

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

G. Dall’Olmo and A. A. Gitelson, “Effect of bio-optical parameter variability on the remote estimation of chlorophyll-a concentration in turbid productive waters: experimental results,” Appl. Opt.44(3), 412–422 (2005).
[CrossRef] [PubMed]

Gleason, A.

Harvey, G. R.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

Hedley, J.

J. Hedley, C. Roelfsema, and S. Phinn, “Efficient radiative transfer model inversion for remote sensing applications,” Remote Sens. Environ.113(11), 2527–2532 (2009).
[CrossRef]

Huang, C.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Kaganovsky, S.

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Kohler, D. D. R.

Korwan, D. R.

Le, C.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Leathers, R. A.

Leavitt, B. C.

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Li, R. R.

Li, Y.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Louchard, E. M.

Lu, H.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Lucke, R. L.

Ma, R.

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

Mahalanobis, P. C.

P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India2(1), 49–55 (1936).

Matsushita, B.

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

McGlothlin, N. R.

Mobley, C. D.

Montes, M. J.

Morel, A.

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr.22(4), 709–722 (1977).
[CrossRef]

Moses, W. J.

W. J. Moses, J. H. Bowles, R. L. Lucke, and M. R. Corson, “Impact of signal-to-noise ratio in a hyperspectral sensor on the accuracy of biophysical parameter estimation in case II waters,” Opt. Express20(4), 4309–4330 (2012).
[CrossRef] [PubMed]

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Ortner, P. B.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

Phinn, S.

J. Hedley, C. Roelfsema, and S. Phinn, “Efficient radiative transfer model inversion for remote sensing applications,” Remote Sens. Environ.113(11), 2527–2532 (2009).
[CrossRef]

Prieur, L.

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr.22(4), 709–722 (1977).
[CrossRef]

Reid, R. P.

Roelfsema, C.

J. Hedley, C. Roelfsema, and S. Phinn, “Efficient radiative transfer model inversion for remote sensing applications,” Remote Sens. Environ.113(11), 2527–2532 (2009).
[CrossRef]

Snyder, W. A.

Steward, R. G.

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

Sulimani, B.

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Sun, D.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Sundman, L. K.

Wood, D. L.

Yacobi, Y. Z.

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Yang, W.

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

Zha, Y.

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Appl. Opt. (4)

IEEE Geosci. Remote Sens. Lett. (1)

W. Yang, B. Matsushita, J. Chen, T. Fukushima, and R. Ma, “An enhanced three-band index for estimating chlorophyll-a in turbid case-II waters: case studies of Lake Kasumigaura, Japan, and Lake Dianchi, China,” IEEE Geosci. Remote Sens. Lett.7(4), 655–659 (2010).
[CrossRef]

Int. J. Remote Sens. (1)

A. Gitelson, “The peak near 700 nm on radiance spectra of algae and water - relationships of its magnitude and position with chlorophyll concentration,” Int. J. Remote Sens.13(17), 3367–3373 (1992).
[CrossRef]

Limnol. Oceanogr. (3)

K. L. Carder, R. G. Steward, G. R. Harvey, and P. B. Ortner, “Marine humic and fulvic acids: their effects on remote sensing of ocean chlorophyll,” Limnol. Oceanogr.34(1), 68–81 (1989).
[CrossRef]

A. Morel and L. Prieur, “Analysis of variations in ocean color,” Limnol. Oceanogr.22(4), 709–722 (1977).
[CrossRef]

C. D. Mobley, “A numerical model for the computation of radiance distributions in natural waters with wind roughened surfaces,” Limnol. Oceanogr.34(8), 1473–1483 (1989).
[CrossRef]

Opt. Express (1)

Proc. Natl. Inst. Sci. India (1)

P. C. Mahalanobis, “On the generalized distance in statistics,” Proc. Natl. Inst. Sci. India2(1), 49–55 (1936).

Remote Sens. Environ. (2)

J. Hedley, C. Roelfsema, and S. Phinn, “Efficient radiative transfer model inversion for remote sensing applications,” Remote Sens. Environ.113(11), 2527–2532 (2009).
[CrossRef]

C. Le, Y. Li, Y. Zha, D. Sun, C. Huang, and H. Lu, “A four-band semi-analytical model for estimating chlorophyll a in highly turbid lakes: the case of Taihu Lake, China,” Remote Sens. Environ.113(6), 1175–1182 (2009).
[CrossRef]

Water Res. (1)

Y. Z. Yacobi, W. J. Moses, S. Kaganovsky, B. Sulimani, B. C. Leavitt, and A. A. Gitelson, “NIR-red reflectance-based algorithms for chlorophyll-a estimation in mesotrophic inland and coastal waters: Lake Kinneret case study,” Water Res.45(7), 2428–2436 (2011).
[CrossRef] [PubMed]

Other (5)

R. P. Bukata, J. H. Jerome, K. Y. Kondratyev, and D. V. Pozdnyakov, Optical Properties and Remote Sensing of Inland and Coastal Waters (CRC Press, 1995).

K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic Press, 2003).

P. Billingsley, Probability and Measure, Third Ed. (John Wiley & Sons, 1995).

C. D. Mobley and L. K. Sundman, Hydrolight 5 Ecolight 5 technical documentation, 1st Ed., (Sequoia Scientific Inc., 2008).

M. J. Montes, B. C. Gao, and C. O. Davis, “A new algorithm for atmospheric correction of hyperspectral remote sensing data,” Proc. SPIE, Geo-Spatial Image and Data Exploitation II, W. E. Roper (ed.), 4383: 23–30 (2001).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

1000 Noisy variations (left) and covariance matrix (right). The latter is scaled by 10−8.

Fig. 2
Fig. 2

Distribution of the retrieved chl-a (left), CDOM (center) and SPM (right) values for the L 2 (blue) and Mahalanobis (red) distances. The true values (indicated by arrow) are 3.0 mg m−3, 0.3 m−1, and 2.0 g m−3, respectively.

Fig. 3
Fig. 3

Distribution of L 2 (left) and Mahalanobis (right) distances from the first noise-free spectrum to the 1000 noisy realizations. The red bars are the actual data, the line represents the fitted normal and chi-squared distributions, respectively.

Fig. 4
Fig. 4

Two-band scattergram of the noisy (left) and database (right) spectra. The red curve represents the 3-sigma range of the L 2 distance; the black dotted curve is the 3-sigma range for MD. The bands shown are 422 and 571 nm.

Fig. 5
Fig. 5

Distribution of the L 2 (left) and Mahalanobis (right) distances in the noise range. The dotted line represents the (fitted) distribution of the noisy data; the bars represent the number of database spectra at that distance. Note that the number of database spectra within the noise range is significantly smaller for the Mahalanobis distance.

Tables (3)

Tables Icon

Table 1 Overview of the range / values for the parameters used in the database search. Some values have been approximated for brevity.

Tables Icon

Table 2 IOP characterizations and experimental results for the 52 noise-free spectra. Cols.1-3 are the input IOPs; Cols. 4-6 describe the retrieval results (out of 1000 noisy realizations); Cols. 7-12 are the avg. retrieved IOP error (see text for definition). Note that the number of correct retrievals will be strongly affected by the discretization of the various parameters in the lookup table (Table 1); as a result, a direct comparison among various levels is difficult.

Tables Icon

Table 3 Distribution of the noisy and database spectra within a given range of the noise-free input spectrum.

Equations (6)

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

d( x,y )= i=1 n ( x i y i ) 2 = ( xy ) t ( xy )
d M ( x )= ( xμ ) t Σ 1 ( xμ )
d M ( x,y )= ( xy ) t Σ 1 ( xy )
d M ( x,y )= i=1 n 1 σ i 2 ( x i y i ) 2
100×( true valueavg. retrieved value true value ).
d( x,ρ )= i=1 n σ i 2 ( 1 σ i 2 ( x i ρ i ) 2 )

Metrics