Abstract

We present the derivation of an analytical model for the self-shading error of an oceanographic upwelling radiometer. The radiometer is assumed to be cylindrical and can either be a profiling instrument or include a wider cylindrical buoy for floating at the sea surface. The model treats both optically shallow and optically deep water conditions and can be applied any distance off the seafloor. We evaluate the model by comparing its results to those from Monte Carlo simulations. The analytical model performs well over a large range of environmental conditions and provides a significant improvement to previous analytical models. The model is intended for investigators who need to apply self-shading corrections to radiometer data but who do not have the ability to compute shading corrections with Monte Carlo simulations. The model also can provide guidance for instrument design and cruise planning.

© 2004 Optical Society of America

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References

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    [CrossRef]
  2. R. A. Leathers, T. V. Downes, and C. D. Mobley, �??Self-shading correction for upwelling sea-surface radiance measurements made with buoyed instruments,�?? Opt. Express 8, 561-570 (2001)
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. J. L. Mueller et al., �??Ocean optics protocols for satellite ocean color sensor validation, Revision 4, Volume III: radiometric measurements and data analysis protocols,�?? National Aeronautical and Space Administration Report 21621 (2003)
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  10. E. Aas and B. Korsbø, �??Self-shading effect by radiance meters on upward radiance observed in coastal waters,�?? Limnol. Oceanogr. 42, 974-980 (1997)
    [CrossRef]
  11. J. Piskozub, A. R. Weeks, J. N. Schwarz, and I. S. Robinson, �??Self-shading of upwelling irradiance for an instrument with sensors on a sidearm,�?? Appl. Opt. 39, 1872-1878 (2000)
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  12. C. D. Mobley, Light and Water (Academic Press, New York, 1994)
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    [CrossRef]
  14. K. S. Baker and R. C. Smith, �??Irradiance transmittance through the air-water interface,�?? in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 556-565 (1990).
    [CrossRef]
  15. J. Piskozub, �??Effects of surface waves and sea-bottom on self-shading on in-water optical instruments,�?? in Ocean Optics XII, J. Jaffe, ed., Proc. SPIE 2258, 300-308 (1994)
    [CrossRef]

Appl. Opt. (3)

Int. J. Remote Sensing (1)

D. R. Lyzenga, �??Remote sensing of bottom reflectance and water attenuation parameters in shallow water using aircraft and Landsat data,�?? Int. J. Remote Sensing 2, 71-82 (1981)
[CrossRef]

J. Geophys. Res. (1)

J. E. O'Reilley, S. Maritorena, B. G. Mitchell, D. A. Siegel, K. L. Carder, S. A. Garver, M. Kahru, and C. McClain, �??Ocean color chlorophyll algorithms for SeaWiFS,�?? J. Geophys. Res. 103, 24937-24953 (1998)
[CrossRef]

Limnol. Oceanogr (1)

E. Aas and B. Korsbø, �??Self-shading effect by radiance meters on upward radiance observed in coastal waters,�?? Limnol. Oceanogr. 42, 974-980 (1997)
[CrossRef]

Limnol. Oceanogr. (3)

E. M. Louchard, R. P. Reid, F. C. Stephens, C. O. Davis, R. A. Leathers, and T. V. Downes, �??Optical remote sensing of benthic habitats and bathymetry in coastal environments at Lee Stocking Island, Bahamas: A comparative spectral classification approach,�?? Limnol. Oceanogr. 48, 511-521 (2003)
[CrossRef]

H. M. Dierssen, R. C. Zimmerman, R. A. Leathers, T. V. Downes, and C. O. Davis, �??Ocean color remote sensing of seagrass and bathymetry in the Bahamas Banks by high-resolution airborne imagery,�?? Limnol. Oceanogr. 48, 444-455 (2003)
[CrossRef]

H. R. Gordon and K. Ding, �??Self-shading of in-water optical instruments,�?? Limnol. Oceanogr. 37, 491-500 (1992)
[CrossRef]

National Aeronautical and Space Administ (1)

J. L. Mueller et al., �??Ocean optics protocols for satellite ocean color sensor validation, Revision 4, Volume III: radiometric measurements and data analysis protocols,�?? National Aeronautical and Space Administration Report 21621 (2003)

Ocean Optics (1)

K. S. Baker and R. C. Smith, �??Irradiance transmittance through the air-water interface,�?? in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE 1302, 556-565 (1990).
[CrossRef]

Ocean Optics XII (1)

J. Piskozub, �??Effects of surface waves and sea-bottom on self-shading on in-water optical instruments,�?? in Ocean Optics XII, J. Jaffe, ed., Proc. SPIE 2258, 300-308 (1994)
[CrossRef]

Opt. Express (2)

Other (1)

C. D. Mobley, Light and Water (Academic Press, New York, 1994)

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Figures (6)

Fig. 1.
Fig. 1.

Shading of an upwelling radiance sensor by a horizontal disk of radius r.

Fig. 2.
Fig. 2.

Model for computing the amount of the overlap between the shadow on the seafloor and the sensor FOV.

Fig. 3.
Fig. 3.

Predictions of Hyper-TSRB self-shading in deep water versus water absorption: Monte Carlo calculations (triangles); analytical model for the Hyper-TSRB (solid line); and analytical model for the sensor head only (dashed lines). Error values are for direct sunlight from the solar zenith angles indicated. Analytical values assume negligible scattering; numerical results are for b = a.

Fig. 4.
Fig. 4.

Predictions of Hyper-TSRB self-shading in deep water versus sun position: Monte Carlo calculations (triangles); analytical model for the Hyper-TSRB (solid line); and analytical model for the sensor head only (dashed lines). Analytical values assume negligible scattering; numerical results are for b = a.

Fig. 5.
Fig. 5.

Hyper-TSRB self-shading error versus scattering coefficient for a = 0.02 m-1. Semi-analytical values (solid lines) were obtained with Eqs. (17) and (18) with k 1 = 0.1, and numerical results (asterisks) were computed with Backward Monte Carlo.

Fig. 6.
Fig. 6.

Hyper-TSRB shading error as predicted with the analytical model compared with MC results (triangles). Results shown are for a = 0.2 m-1, b = 0.4 m-1, R b = 0.2, FOV half-angle = 20°, and the indicated values of the solar zenith angle. The indicated depth is the total water depth; the sensor is 0.66 m below the sea surface.

Tables (1)

Tables Icon

Table 1. Values of k [Eq. (9)] for a radiance point sensor

Equations (21)

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z 0 = r tan θ 0 w ,
θ 0 w = sin 1 ( sin θ 0 1.338 ) .
L u t ( z ) = L u t ( z s ) exp [ K u ( z z s ) ] ,
L u m = L u t ( z 0 ) exp [ a ( z 0 z s ) ] .
L u m = L u t ( z s ) exp [ ( K u + a ) ( z 0 z s ) ] .
L u m = L u t ( z s ) exp [ k a ( r z s tan θ 0 w ) ] , tan θ 0 w < r z s ,
k = ( 1 tan θ 0 w + 1 sin θ 0 w ) .
ε w = 1 L u m L u t ( z s ) = { 1 exp [ k a ( r z s tan θ 0 w ) ] , tan θ 0 w < r z s 0 , tan θ 0 w > r z s } .
ε w = [ 1 exp ( k a r ) ] .
ε w = { 1 exp [ k a ( r b z s tan θ 0 w ) ] , tan θ 0 w < ( r s r b ) z s 1 exp [ k a r s ] , tan θ 0 w > ( r s r b ) z s } .
ε B = { 0 , ( x d r d ) > r fov 1 , ( r fov ) > ( x d r d ) ( r d r fov ) 2 , ( x d + r d ) > r fov 2 π r fov 2 ( x d r d x int r d 2 ( x x d ) 2 dx + x int r fov r fov 2 x 2 dx ) , { ( r fov ) < ( x d r d ) < r fov and r fov < ( x d + r d ) } } ,
x d tan θ 0 w ( z bot z d ) , r fov = [ tan θ fov ( z bot z s ) ] , and x int = r fov 2 r d 2 + x d 2 2 x d .
x d r d x int r d 2 ( x x d ) 2 dx = x int x d 2 r d 2 ( x int x d ) 2 + r d 2 2 sin 1 ( x int x d r d ) + π r d 2 4
x int r fov r fov 2 x 2 dx = π r fov 2 4 x int 2 r fov 2 x int 2 r fov 2 2 sin 1 ( x int r fov ) .
ε B = max ( ε s , ε b ) ,
ε = ( L uw ( z s ) L u t ( z s ) ) ε w + ( L uB ( z s ) L u t ( z s ) ) ε B ,
L uw ( z s ) L u t ( z s ) = b b { 1 exp [ a χ z d ] } b b + ( R b a μ 0 w χ b b ) exp [ a χ z d ] ,
ε = i w i ε i ,
ε = ε sky f + ( 1 f ) ε sun ,
ε = 1 exp [ ( 1 + 1 cos θ 0 w ) a ( z 0 z s ) ] ,
z 0 = r d exp ( k 1 b z 0 ) ,

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