Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: Theoretical development and calculations

Nicholas R. Nalli; Peter J. Minnett; Paul van Delst

doi:10.1364/AO.47.003701

Applied Optics
Vol. 47,
Issue 21,
pp. 3701-3721
(2008)
•https://doi.org/10.1364/AO.47.003701

Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: Theoretical development and calculations

Nicholas R. Nalli, Peter J. Minnett, and Paul van Delst

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Nicholas R. Nalli, Peter J. Minnett, and Paul van Delst, "Emissivity and reflection model for calculating unpolarized isotropic water surface-leaving radiance in the infrared. I: Theoretical development and calculations," Appl. Opt. 47, 3701-3721 (2008)

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Abstract

Although published sea surface infrared (IR) emissivity models have gained widespread acceptance for remote sensing applications, discrepancies have been identified against field observations obtained from IR Fourier transform spectrometers at view angles $≳ 40 °$ . We therefore propose, in this two-part paper, an alternative approach for calculating surface-leaving IR radiance that treats both emissivity and atmospheric reflection in a systematic yet practical manner. This first part presents the theoretical basis, development, and computations of the proposed model.

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	Wind Speed, ${\bar{U}}_{10}$ ( ${ms}^{- 1}$ )
	Cox and Munk [15] PDF						Ebuchi and Kizu [38] PDF
$θ_{0}$	0	4	8	12	16	20	0	4	8	12	16	20
0	2.8	7.8	10.5	12.6	14.4	15.9	7.1	9.7	11.6	13.2	14.6	15.9
8	8.3	10.6	12.6	14.4	15.9	17.2	10.2	12.0	13.5	14.9	16.1	17.2
16	16.1	17.0	18.0	19.0	20.0	20.9	16.9	17.7	18.5	19.4	20.1	20.9
24	24.1	24.5	24.9	25.4	25.8	26.3	24.4	24.8	25.1	25.5	25.9	26.3
32	32.0	32.1	32.2	32.4	32.5	32.6	32.1	32.2	32.3	32.4	32.5	32.6
40	40.0	39.8	39.7	39.6	39.5	39.4	39.9	39.8	39.6	39.5	39.4	39.4
48	47.9	47.6	47.2	46.8	46.5	46.1	47.6	47.3	47.0	46.7	46.4	46.1
52.5	52.4	51.9	51.4	50.9	50.4	49.9	52.0	51.5	51.1	50.7	50.3	49.9
56.5	56.4	55.7	55.0	54.4	53.7	53.1	55.8	55.3	54.7	54.2	53.6	53.1
60.5	60.4	59.5	58.7	57.8	57.0	56.3	59.7	58.9	58.2	57.6	56.9	56.3
64.5	64.3	63.3	62.2	61.2	60.2	59.3	63.4	62.5	61.7	60.8	60.1	59.3
68.5	68.3	66.9	65.6	64.4	63.2	62.2	67.2	66.0	65.0	64.0	63.0	62.2
72.5	72.2	70.5	68.8	67.4	66.0	64.8	70.8	69.4	68.1	66.9	65.8	64.8
76.5	76.2	73.8	71.8	70.1	68.6	67.3	74.2	72.5	70.9	69.6	68.4	67.3
80.5	80.0	76.9	74.5	72.6	71.0	69.7	77.3	75.2	73.5	72.1	70.8	69.7
84.5	83.6	79.5	76.9	74.9	73.2	71.8	80.0	77.7	75.8	74.3	73.0	71.8
88.5	86.5	81.6	78.9	76.9	75.2	73.8	82.2	79.7	77.8	76.3	74.9	73.8
92.5	88.1	83.3	80.6	78.6	77.0	75.6	83.9	81.4	79.6	78.0	76.7	75.6
96.5	88.8	84.7	82.1	80.2	78.6	77.2	85.2	82.9	81.1	79.6	78.4	77.2
100.5	90.0	85.7	83.4	81.6	80.1	78.8	86.2	84.1	82.4	81.1	79.9	78.8
104.5	NAN	86.5	84.4	82.7	81.4	80.2	86.9	85.0	83.6	82.3	81.2	80.2
108.5	NAN	87.2	85.4	83.9	82.6	81.5	87.6	85.9	84.6	83.4	82.4	81.5
112.5	NAN	87.9	86.0	84.8	83.6	82.7	88.4	86.6	85.5	84.5	83.4	82.7
116.5	NAN	89.2	86.8	85.7	84.7	83.7	NAN	87.3	86.1	85.2	84.5	83.7
120.5	NAN	NAN	87.6	86.4	85.5	84.6	NAN	88.3	86.9	86.2	85.4	84.6

	Wind Speed, ${\bar{U}}_{10}$ ( ${ms}^{- 1}$ )
	Cox and Munk [15] PDF						Ebuchi and Kizu [38] PDF
$θ_{0}$	0	4	8	12	16	20	0	4	8	12	16	20
0	5.6	15.6	21.3	25.6	29.3	32.5	14.4	19.5	23.5	26.9	29.8	32.5
8	9.4	17.1	22.4	26.5	30.1	33.2	16	20.8	24.5	27.8	30.6	33.2
16	16.6	21.3	25.6	29.2	32.4	35.2	20.5	24.2	27.4	30.3	32.8	35.2
24	24.4	27.3	30.4	33.3	36.0	38.5	26.8	29.4	31.9	34.2	36.4	38.5
32	32.3	34.3	36.5	38.6	40.8	42.8	33.9	35.7	37.6	39.4	41.1	42.8
40	40.2	41.7	43.2	44.8	46.4	48.1	41.4	42.7	44.0	45.4	46.7	48.0
48	48.2	49.2	50.4	51.5	52.7	54.0	49.1	50.0	51.0	51.9	52.9	54.0
52.5	52.6	53.6	54.5	55.5	56.5	57.5	53.4	54.2	55.0	55.8	56.6	57.5
56.5	56.6	57.4	58.2	59.0	59.8	60.6	57.3	57.9	58.6	59.3	60.0	60.6
60.5	60.6	61.3	62.0	62.6	63.2	63.6	61.2	61.7	62.3	62.8	63.3	63.6
64.5	64.6	65.2	65.7	66.0	66.2	66.3	65.1	65.5	65.9	66.1	66.2	66.3
68.5	68.6	69.0	69.3	69.2	68.8	68.5	69.0	69.3	69.3	69.2	68.8	68.5
72.5	72.6	72.9	72.5	71.8	70.9	70.3	72.8	72.7	72.3	71.5	70.8	70.3
76.5	76.5	76.4	75.2	73.6	72.5	71.0	76.5	75.6	74.4	73.2	72.3	71.0
80.5	80.5	79.0	76.7	74.8	73.1	71.3	79.6	77.6	75.7	74.3	72.7	71.3
84.5	84.4	80.4	77.5	75.1	72.8	71.0	81.3	78.3	76.3	74.2	72.5	71.0
88.5	86.7	80.5	77.0	74.3	72.2	70.4	81.3	78.0	75.6	73.6	71.9	70.4
92.5	85.3	79.3	75.7	73.0	71.1	69.3	80.1	76.8	74.3	72.3	70.8	69.3
96.5	82.3	77.2	74.3	71.5	69.3	67.5	78.0	75.2	72.8	70.7	68.9	67.5
100.5	79.4	74.5	71.4	69.8	67.5	65.6	75.5	72.7	70.9	69.0	67.2	65.6
104.5	NAN	71.6	69.5	67.4	65.6	63.6	72.1	69.3	67.8	66.9	65.2	63.6
108.5	NAN	68.7	66.0	64.3	63.3	61.5	69.0	67.4	65.7	63.3	63.0	61.5
112.5	NAN	65.1	63.1	62.1	60.3	59.3	65.9	64.2	62.1	61.2	60.5	59.3
116.5	NAN	62.3	60.4	58.3	57.3	56.9	NAN	60.8	59.7	58.7	56.8	56.9
120.5	NAN	NAN	56.5	56.0	55.0	53.2	NAN	57.7	56.5	55.2	54.7	53.1

	Wind Speed, ${\bar{U}}_{10}$ ( ${ms}^{- 1}$ )
$θ_{0}$	0	2	4	5	8	10	12	14	16	18	20
Cox and Munk [15] PDF
15	13.0	0.1	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
20	19.6	17.8	14.6	7.1	2.5	1.8	1.7	1.4	0.2	0.0	0.0
25	25.0	24.7	24.4	24.0	23.4	22.6	21.2	18.6	11.3	9.3	6.2
30	30.1	30.3	30.4	30.4	30.3	30.1	29.7	29.3	28.6	27.8	26.6
35	35.1	35.4	35.6	35.7	35.7	35.6	35.4	35.1	34.7	34.2	33.7
40	40.1	40.4	40.6	40.6	40.6	40.5	40.3	40.0	39.8	39.4	39.1
45	45.1	45.3	45.4	45.4	45.2	45.1	44.8	44.6	44.3	43.9	43.6
50	50.0	50.1	50.1	49.9	49.7	49.4	49.1	48.8	48.5	48.2	47.9
55	55.0	54.9	54.6	54.4	54.0	53.7	53.4	53.1	52.8	52.4	52.1
60	59.9	59.6	59.1	58.7	58.3	57.9	57.6	57.2	56.8	56.5	56.2
65	64.8	64.2	63.5	63.1	62.5	62.1	61.7	61.3	61.0	60.7	60.4
70	69.7	68.7	68.1	67.5	67.0	66.6	66.2	65.8	65.4	65.1	64.8
Ebuchi and Kizu [38] PDF
15	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
20	16.1	8.5	6.5	2.3	1.8	1.7	1.5	1.1	0.0	0.0	0.0
25	24.5	24.3	23.9	23.3	22.6	21.5	19.7	13.4	10.0	9.1	6.2
30	30.3	30.4	30.4	30.3	30.1	29.8	29.4	29.0	28.3	27.6	26.6
35	35.5	35.7	35.7	35.7	35.6	35.5	35.2	34.9	34.5	34.1	33.7
40	40.5	40.6	40.7	40.6	40.5	40.3	40.1	39.9	39.6	39.4	39.1
45	45.4	45.4	45.3	45.2	45.1	44.9	44.7	44.4	44.2	43.9	43.6
50	50.1	50.0	49.8	49.6	49.4	49.2	48.9	48.7	48.4	48.2	47.9
55	54.7	54.5	54.3	54.0	53.7	53.4	53.2	52.9	52.7	52.4	52.1
60	59.3	58.9	58.5	58.3	57.9	57.6	57.3	57.0	56.7	56.4	56.2
65	63.7	63.3	62.9	62.5	62.1	61.8	61.5	61.2	60.9	60.6	60.4
70	68.3	67.7	67.3	66.9	66.6	66.2	65.9	65.6	65.3	65.1	64.8

	Wind Speed, ${\bar{U}}_{10}$ ( ${ms}^{- 1}$ )
	Cox and Munk [15] PDF						Ebuchi and Kizu [38] PDF
$θ_{0}$	0	4	8	12	16	20	0	4	8	12	16	20
0	2.8	7.8	10.5	12.6	14.4	15.9	7.1	9.7	11.6	13.2	14.6	15.9
8	8.3	10.6	12.6	14.4	15.9	17.2	10.2	12.0	13.5	14.9	16.1	17.2
16	16.1	17.0	18.0	19.0	20.0	20.9	16.9	17.7	18.5	19.4	20.1	20.9
24	24.1	24.5	24.9	25.4	25.8	26.3	24.4	24.8	25.1	25.5	25.9	26.3
32	32.0	32.1	32.2	32.4	32.5	32.6	32.1	32.2	32.3	32.4	32.5	32.6
40	40.0	39.8	39.7	39.6	39.5	39.4	39.9	39.8	39.6	39.5	39.4	39.4
48	47.9	47.6	47.2	46.8	46.5	46.1	47.6	47.3	47.0	46.7	46.4	46.1
52.5	52.4	51.9	51.4	50.9	50.4	49.9	52.0	51.5	51.1	50.7	50.3	49.9
56.5	56.4	55.7	55.0	54.4	53.7	53.1	55.8	55.3	54.7	54.2	53.6	53.1
60.5	60.4	59.5	58.7	57.8	57.0	56.3	59.7	58.9	58.2	57.6	56.9	56.3
64.5	64.3	63.3	62.2	61.2	60.2	59.3	63.4	62.5	61.7	60.8	60.1	59.3
68.5	68.3	66.9	65.6	64.4	63.2	62.2	67.2	66.0	65.0	64.0	63.0	62.2
72.5	72.2	70.5	68.8	67.4	66.0	64.8	70.8	69.4	68.1	66.9	65.8	64.8
76.5	76.2	73.8	71.8	70.1	68.6	67.3	74.2	72.5	70.9	69.6	68.4	67.3
80.5	80.0	76.9	74.5	72.6	71.0	69.7	77.3	75.2	73.5	72.1	70.8	69.7
84.5	83.6	79.5	76.9	74.9	73.2	71.8	80.0	77.7	75.8	74.3	73.0	71.8
88.5	86.5	81.6	78.9	76.9	75.2	73.8	82.2	79.7	77.8	76.3	74.9	73.8
92.5	88.1	83.3	80.6	78.6	77.0	75.6	83.9	81.4	79.6	78.0	76.7	75.6
96.5	88.8	84.7	82.1	80.2	78.6	77.2	85.2	82.9	81.1	79.6	78.4	77.2
100.5	90.0	85.7	83.4	81.6	80.1	78.8	86.2	84.1	82.4	81.1	79.9	78.8
104.5	NAN	86.5	84.4	82.7	81.4	80.2	86.9	85.0	83.6	82.3	81.2	80.2
108.5	NAN	87.2	85.4	83.9	82.6	81.5	87.6	85.9	84.6	83.4	82.4	81.5
112.5	NAN	87.9	86.0	84.8	83.6	82.7	88.4	86.6	85.5	84.5	83.4	82.7
116.5	NAN	89.2	86.8	85.7	84.7	83.7	NAN	87.3	86.1	85.2	84.5	83.7
120.5	NAN	NAN	87.6	86.4	85.5	84.6	NAN	88.3	86.9	86.2	85.4	84.6

	Wind Speed, ${\bar{U}}_{10}$ ( ${ms}^{- 1}$ )
	Cox and Munk [15] PDF						Ebuchi and Kizu [38] PDF
$θ_{0}$	0	4	8	12	16	20	0	4	8	12	16	20
0	5.6	15.6	21.3	25.6	29.3	32.5	14.4	19.5	23.5	26.9	29.8	32.5
8	9.4	17.1	22.4	26.5	30.1	33.2	16	20.8	24.5	27.8	30.6	33.2
16	16.6	21.3	25.6	29.2	32.4	35.2	20.5	24.2	27.4	30.3	32.8	35.2
24	24.4	27.3	30.4	33.3	36.0	38.5	26.8	29.4	31.9	34.2	36.4	38.5
32	32.3	34.3	36.5	38.6	40.8	42.8	33.9	35.7	37.6	39.4	41.1	42.8
40	40.2	41.7	43.2	44.8	46.4	48.1	41.4	42.7	44.0	45.4	46.7	48.0
48	48.2	49.2	50.4	51.5	52.7	54.0	49.1	50.0	51.0	51.9	52.9	54.0
52.5	52.6	53.6	54.5	55.5	56.5	57.5	53.4	54.2	55.0	55.8	56.6	57.5
56.5	56.6	57.4	58.2	59.0	59.8	60.6	57.3	57.9	58.6	59.3	60.0	60.6
60.5	60.6	61.3	62.0	62.6	63.2	63.6	61.2	61.7	62.3	62.8	63.3	63.6
64.5	64.6	65.2	65.7	66.0	66.2	66.3	65.1	65.5	65.9	66.1	66.2	66.3
68.5	68.6	69.0	69.3	69.2	68.8	68.5	69.0	69.3	69.3	69.2	68.8	68.5
72.5	72.6	72.9	72.5	71.8	70.9	70.3	72.8	72.7	72.3	71.5	70.8	70.3
76.5	76.5	76.4	75.2	73.6	72.5	71.0	76.5	75.6	74.4	73.2	72.3	71.0
80.5	80.5	79.0	76.7	74.8	73.1	71.3	79.6	77.6	75.7	74.3	72.7	71.3
84.5	84.4	80.4	77.5	75.1	72.8	71.0	81.3	78.3	76.3	74.2	72.5	71.0
88.5	86.7	80.5	77.0	74.3	72.2	70.4	81.3	78.0	75.6	73.6	71.9	70.4
92.5	85.3	79.3	75.7	73.0	71.1	69.3	80.1	76.8	74.3	72.3	70.8	69.3
96.5	82.3	77.2	74.3	71.5	69.3	67.5	78.0	75.2	72.8	70.7	68.9	67.5
100.5	79.4	74.5	71.4	69.8	67.5	65.6	75.5	72.7	70.9	69.0	67.2	65.6
104.5	NAN	71.6	69.5	67.4	65.6	63.6	72.1	69.3	67.8	66.9	65.2	63.6
108.5	NAN	68.7	66.0	64.3	63.3	61.5	69.0	67.4	65.7	63.3	63.0	61.5
112.5	NAN	65.1	63.1	62.1	60.3	59.3	65.9	64.2	62.1	61.2	60.5	59.3
116.5	NAN	62.3	60.4	58.3	57.3	56.9	NAN	60.8	59.7	58.7	56.8	56.9
120.5	NAN	NAN	56.5	56.0	55.0	53.2	NAN	57.7	56.5	55.2	54.7	53.1

Abstract

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Figures (15)

Tables (3)

Equations (37)

Applied Optics