Abstract

The spectral distribution of upwelling and downwelling irradiance were used to estimate the effective upwelling irradiance depth as well as examine the angular distribution of the downwelling radiance. The effective upwelling depth was seen to undergo spectral “shifts” in wavelength maxima in relation to elevated particulate concentrations. Wavelengths of the UVA minimum and mid visible maximum depths were found to be shifted to higher wavelengths (red shifted) at high particulate concentrations, while expected minimums at chlorophyll and phycocyanin absorption peaks and in the NIR were shifted to lower wavelengths (blue shifted). By comparing upwelling and downwelling irradiance profiles, the wavelength limits of the asymptotic angular radiance distribution were found to correspond to the visible spectral domain (390 – 740 nm).

©2011 Optical Society of America

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References

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    [Crossref]
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    [Crossref]

2010 (3)

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

A. E. Hickman, S. Dutkiewicz, R. G. Williams, and M. J. Follows, “Modelling the effects of chromatic adaptation on phytoplankton community structure in the oligotrophic ocean,” Mar. Ecol. Prog. Ser. 406, 1–17 (2010).
[Crossref]

2009 (3)

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48(35), 6811–6819 (2009).
[Crossref] [PubMed]

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

2008 (1)

R. Ma, H. Duan, X. Gu, and S. Zhang, “Detecting aquatic vegetation changes in Taihu Lake, China using multi-temporal satellite imagery,” Sensors 8(6), 3988–4005 (2008).
[Crossref]

2007 (2)

2006 (2)

M. Chami, D. McKee, E. Leymarie, and G. Khomenko, “Influence of the angular shape of the volume-scattering function and multiple scattering on remote sensing reflectance,” Appl. Opt. 45(36), 9210–9220 (2006).
[Crossref] [PubMed]

R. Ma, J. Tang, and J. Dai, “Bio-optical model with optimal parameter suitable for Taihu Lake in water colour remote sensing,” Int. J. Remote Sens. 27(19), 4305–4328 (2006).
[Crossref]

2004 (3)

R. Yan, F. Kong, and X. Han, “Analysis of the recruitment of the winter survival algae on the sediments of Lake Taihu by fluorometry,” J. Lake. Sci. 16, 163–168(in Chinese with English abstract) (2004).

D. G. Bowers, D. Evans, and D. N. Thomas, “Interpreting the colour of an estuary,” Estuar. Coast. Shelf Sci. 59(1), 13–20 (2004).
[Crossref]

A. Islam, J. Gao, and W. Ahmad, “A composite DOP approach to excluding bottom reflectance in mapping water parameters of shallow coastal zones from TM imagery,” Remote Sens. Environ. 92(1), 40–51 (2004).
[Crossref]

2002 (1)

M. M. Squires, L. F. W. Lesack, and D. Huebert, “The influence of water transparency on the distribution and abundance of macrophytes among lakes of the Mackenzie Delta, Western Canadian Arctic,” Freshw. Biol. 47(11), 2123–2135 (2002).
[Crossref]

2001 (1)

T. Ohde and H. Siegel, “Correction of bottom influence in ocean colour satellite images of shallow water areas of the Baltic Sea,” Int. J. Remote Sens. 22(2), 297–313 (2001).
[Crossref]

1998 (2)

Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, J. S. Patch, and B. Gentili, “Hyperspectral remote sensing for shallow waters. I. A semianalytical model,” Appl. Opt. 37(27), 6329–6338 (1998).
[Crossref]

R. A. Maffione, “Theoretical developments on the optical properties of highly turbid waters and sea ice,” Limnol. Oceanogr. 43(1), 29–33 (1998).
[Crossref]

1989 (1)

J. T. O. Kirk, “The Upwelling Light Stream in Natural Waters,” Limnol. Oceanogr. 34(8), 1410–1425 (1989).
[Crossref]

1986 (1)

B. J. Topliss, “Spectral variations in upwelling radiant intensity in turbid coastal waters,” Estuar. Coast. Shelf Sci. 22(4), 395–414 (1986).
[Crossref]

Ahmad, W.

A. Islam, J. Gao, and W. Ahmad, “A composite DOP approach to excluding bottom reflectance in mapping water parameters of shallow coastal zones from TM imagery,” Remote Sens. Environ. 92(1), 40–51 (2004).
[Crossref]

Bergamino, N.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Bowers, D. G.

D. G. Bowers, D. Evans, and D. N. Thomas, “Interpreting the colour of an estuary,” Estuar. Coast. Shelf Sci. 59(1), 13–20 (2004).
[Crossref]

Bracchini, L.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Bricaud, A.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Carder, K. L.

Chami, M.

Chang, C. H.

Charriere, B.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Cornet, Y.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Dai, J.

R. Ma, J. Tang, and J. Dai, “Bio-optical model with optimal parameter suitable for Taihu Lake in water colour remote sensing,” Int. J. Remote Sens. 27(19), 4305–4328 (2006).
[Crossref]

Dattilo, A. M.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Descy, J. P.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Duan, H.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

R. Ma, H. Duan, X. Gu, and S. Zhang, “Detecting aquatic vegetation changes in Taihu Lake, China using multi-temporal satellite imagery,” Sensors 8(6), 3988–4005 (2008).
[Crossref]

Dutkiewicz, S.

A. E. Hickman, S. Dutkiewicz, R. G. Williams, and M. J. Follows, “Modelling the effects of chromatic adaptation on phytoplankton community structure in the oligotrophic ocean,” Mar. Ecol. Prog. Ser. 406, 1–17 (2010).
[Crossref]

Evans, D.

D. G. Bowers, D. Evans, and D. N. Thomas, “Interpreting the colour of an estuary,” Estuar. Coast. Shelf Sci. 59(1), 13–20 (2004).
[Crossref]

Follows, M. J.

A. E. Hickman, S. Dutkiewicz, R. G. Williams, and M. J. Follows, “Modelling the effects of chromatic adaptation on phytoplankton community structure in the oligotrophic ocean,” Mar. Ecol. Prog. Ser. 406, 1–17 (2010).
[Crossref]

Gao, J.

A. Islam, J. Gao, and W. Ahmad, “A composite DOP approach to excluding bottom reflectance in mapping water parameters of shallow coastal zones from TM imagery,” Remote Sens. Environ. 92(1), 40–51 (2004).
[Crossref]

Gentili, B.

Gu, X.

R. Ma, H. Duan, X. Gu, and S. Zhang, “Detecting aquatic vegetation changes in Taihu Lake, China using multi-temporal satellite imagery,” Sensors 8(6), 3988–4005 (2008).
[Crossref]

Guo, L.

L. Guo, “Ecology. Doing battle with the green monster of Taihu Lake,” Science 317(5842), 1166 (2007).
[Crossref] [PubMed]

Han, X.

R. Yan, F. Kong, and X. Han, “Analysis of the recruitment of the winter survival algae on the sediments of Lake Taihu by fluorometry,” J. Lake. Sci. 16, 163–168(in Chinese with English abstract) (2004).

Hao, J.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

Hickman, A. E.

A. E. Hickman, S. Dutkiewicz, R. G. Williams, and M. J. Follows, “Modelling the effects of chromatic adaptation on phytoplankton community structure in the oligotrophic ocean,” Mar. Ecol. Prog. Ser. 406, 1–17 (2010).
[Crossref]

Horion, S.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Huebert, D.

M. M. Squires, L. F. W. Lesack, and D. Huebert, “The influence of water transparency on the distribution and abundance of macrophytes among lakes of the Mackenzie Delta, Western Canadian Arctic,” Freshw. Biol. 47(11), 2123–2135 (2002).
[Crossref]

Hull, V.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Islam, A.

A. Islam, J. Gao, and W. Ahmad, “A composite DOP approach to excluding bottom reflectance in mapping water parameters of shallow coastal zones from TM imagery,” Remote Sens. Environ. 92(1), 40–51 (2004).
[Crossref]

Khomenko, G.

Kirk, J. T. O.

J. T. O. Kirk, “The Upwelling Light Stream in Natural Waters,” Limnol. Oceanogr. 34(8), 1410–1425 (1989).
[Crossref]

Kong, F.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

R. Yan, F. Kong, and X. Han, “Analysis of the recruitment of the winter survival algae on the sediments of Lake Taihu by fluorometry,” J. Lake. Sci. 16, 163–168(in Chinese with English abstract) (2004).

Kong, W.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

Lee, Z.

Lesack, L. F. W.

M. M. Squires, L. F. W. Lesack, and D. Huebert, “The influence of water transparency on the distribution and abundance of macrophytes among lakes of the Mackenzie Delta, Western Canadian Arctic,” Freshw. Biol. 47(11), 2123–2135 (2002).
[Crossref]

Leymarie, E.

Liu, C.-C.

Loiselle, S.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Loiselle, S. A.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Ma, R.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

R. Ma, H. Duan, X. Gu, and S. Zhang, “Detecting aquatic vegetation changes in Taihu Lake, China using multi-temporal satellite imagery,” Sensors 8(6), 3988–4005 (2008).
[Crossref]

R. Ma, J. Tang, and J. Dai, “Bio-optical model with optimal parameter suitable for Taihu Lake in water colour remote sensing,” Int. J. Remote Sens. 27(19), 4305–4328 (2006).
[Crossref]

Maffione, R. A.

R. A. Maffione, “Theoretical developments on the optical properties of highly turbid waters and sea ice,” Limnol. Oceanogr. 43(1), 29–33 (1998).
[Crossref]

McKee, D.

Mobley, C. D.

Ohde, T.

T. Ohde and H. Siegel, “Correction of bottom influence in ocean colour satellite images of shallow water areas of the Baltic Sea,” Int. J. Remote Sens. 22(2), 297–313 (2001).
[Crossref]

Para, J.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Patch, J. S.

Plisnier, P. D.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Raimbault, P.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Rossi, C.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Sempéré, R.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Shang, L.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

Siegel, H.

T. Ohde and H. Siegel, “Correction of bottom influence in ocean colour satellite images of shallow water areas of the Baltic Sea,” Int. J. Remote Sens. 22(2), 297–313 (2001).
[Crossref]

Squires, M. M.

M. M. Squires, L. F. W. Lesack, and D. Huebert, “The influence of water transparency on the distribution and abundance of macrophytes among lakes of the Mackenzie Delta, Western Canadian Arctic,” Freshw. Biol. 47(11), 2123–2135 (2002).
[Crossref]

Stenuite, S.

N. Bergamino, S. Horion, S. Stenuite, Y. Cornet, S. Loiselle, P. D. Plisnier, and J. P. Descy, “Spatio-temporal dynamics of phytoplankton and primary production in Lake Tanganyika using a MODIS based bio-optical time series,” Remote Sens. Environ. 114(4), 772–780 (2010).
[Crossref]

Steward, R. G.

Sullivan, J. M.

Tang, J.

R. Ma, J. Tang, and J. Dai, “Bio-optical model with optimal parameter suitable for Taihu Lake in water colour remote sensing,” Int. J. Remote Sens. 27(19), 4305–4328 (2006).
[Crossref]

Tedetti, M.

M. Tedetti, B. Charriere, A. Bricaud, J. Para, P. Raimbault, and R. Sempéré, “Distribution of normalized water- leaving radiances at UV and visible wave bands in relation with chlorophyll a and colored detrital matter content in the southeast Pacific,” J. Geophys. Res. 115(C2), C02010 (2010).
[Crossref]

Thomas, D. N.

D. G. Bowers, D. Evans, and D. N. Thomas, “Interpreting the colour of an estuary,” Estuar. Coast. Shelf Sci. 59(1), 13–20 (2004).
[Crossref]

Tognazzi, A.

L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

Topliss, B. J.

B. J. Topliss, “Spectral variations in upwelling radiant intensity in turbid coastal waters,” Estuar. Coast. Shelf Sci. 22(4), 395–414 (1986).
[Crossref]

Twardowski, M. S.

Wen, C. G.

Williams, R. G.

A. E. Hickman, S. Dutkiewicz, R. G. Williams, and M. J. Follows, “Modelling the effects of chromatic adaptation on phytoplankton community structure in the oligotrophic ocean,” Mar. Ecol. Prog. Ser. 406, 1–17 (2010).
[Crossref]

Xu, X.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

Yan, R.

R. Yan, F. Kong, and X. Han, “Analysis of the recruitment of the winter survival algae on the sediments of Lake Taihu by fluorometry,” J. Lake. Sci. 16, 163–168(in Chinese with English abstract) (2004).

Zhang, S.

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
[Crossref] [PubMed]

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[Crossref]

Appl. Opt. (3)

Environ. Sci. Technol. (1)

H. Duan, R. Ma, X. Xu, F. Kong, S. Zhang, W. Kong, J. Hao, and L. Shang, “Two-decade reconstruction of algal blooms in China’s Lake Taihu,” Environ. Sci. Technol. 43(10), 3522–3528 (2009).
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L. Bracchini, A. M. Dattilo, V. Hull, S. A. Loiselle, A. Tognazzi, and C. Rossi, “Modelling Upwelling Irradiance using Secchi disk depth in lake ecosystems,” J. Limnol. 68, 83–91 (2009).

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Figures (5)

Fig. 1
Fig. 1

(a) Non linear fitting of the upwelling and downwelling irradiance at one site in Taihu Lake. (b) Spectral distribution of vertical attenuation coefficients of upwelling and downwelling diffuse irradiance from three typical sites of Taihu Lake. Site 32, located on the eastern part of the lake has a spectral attenuation (Kd(λ) and Ku(λ)) that is close to the average of all sites (at 500 nm). Site 12, located in the north part of the lake between Meiliang and Gonghu Bays has a higher attenuation (one standard deviation larger at 500 nm). Site 78, located on the western part of the lake is characterized by lower attenuation (close to one standard deviation smaller at 500 nm).

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Fig. 2
Fig. 2

(a) The spectral irradiance reflectance (R(λ)) from three sites in Taihu Lake, (b) the spectral irradiance reflectance from different depths from site 32 in Taihu Lake, a site with average attenuation.

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Fig. 3
Fig. 3

The percentage of total upwelling irradiance measured just below the water’s surface which originating at or below 0.30 m (a) and 0.60 m (b).

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Fig. 4
Fig. 4

Spectral distribution of effective upwelling depth from three sites in Taihu Lake. Water surface (dashed line) also shown. The spectral range of asymptotic radiance differ between sites, site 12 (400 – 701 nm), site 32 (380 – 790 nm), site 78 (320 – 795 nm).

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Fig. 5
Fig. 5

Effective depth for Taihu Lake in October 2008 at (a) 412 nm, (b) 442 nm, (c) 559 nm, (d) 683 nm. Sample sites are indicated by a red circle, sites 12, 32 and 78 indicated. It should be noted that it was not possible to make measurements in the shallow bay in the southeast part of the lake (in white).

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Tables (1)

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Table 1 Pearson Correlation Coefficients (r) Between Peaks (Wavelength, Peak Value) of Effective Upwelling Depth and CDOM Absorption, Total Suspended Particulate Matter Concentrations, Chlorophyll a and Phycocyanin Concentrations

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Equations (5)

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E u ( λ , z ) = E u ( λ , 0 )     e − K u ( λ )     z   ( W   m − 2 )
E u ( λ , 0 ) ' = E u ( λ , 0.60 )     e − κ ( λ )     0.60   ( W   m − 2 ) .
E u ( λ , 0 ) = ∫ z e f f e c t i v e ( λ ) 0 E u ( λ , 0 )     e − K u ( λ )     z   e − κ ( λ )     z   d z   ( W   m − 2 )
P ( λ ) = E u ( λ , 0 ) ' E u ( λ , 0 ) = E u ( λ , z ' ) e − κ ( λ )     z ' E u ( λ , 0 ) =   E u ( λ , 0 ) e − K u ( λ )   z '   e − κ ( λ )     z ' E u ( λ , 0 ) P ( λ ) = e − z '   ( K u ( λ )   + κ ( λ ) )
ln ( 0.01 ) = − ( K u ( λ )   + κ ( λ ) ) ( z e f f e c t i v e ( λ ) ) z e f f e c t i v e ( λ ) = 4.61 K u ( λ )   + κ ( λ )

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