Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces

C. L. Walthall, J. M. Norman, J. M. Welles, G. Campbell, and B. L. Blad

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C. L. Walthall, J. M. Norman, J. M. Welles, G. Campbell, and B. L. Blad, "Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces," Appl. Opt. 24, 383-387 (1985)

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Abstract

A simple equation has been developed for describing the bidirectional reflectance of some vegetative canopies and bare soil surfaces. The equation describes directional reflectance as a function of zenith and azimuth view angles and solar azimuth angle. The equation works for simulated and field measured red and IR reflectance under clear sky conditions. Hemispherical reflectance can be calculated as a function of the simple equation coefficients by integrating the equation over the hemisphere of view angles. A single equation for estimating soil bidirectional reflectance was obtained using the relationships between solar zenith angles and the simple equation coefficients for medium and rough soil distributions. The equation has many useful applications such as providing a lower level boundary condition in complex plant canopy models and providing an additional tool for studying bidirectional effects on pointable sensors.

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Canopy	Solar Zenith (deg)	Visible Reflectance Coefficients			RMS Visible (%)	Near Infrared Reflectance Coefficients			RMS Near Infrared (%)
Canopy	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Visible (%)	a (%/rad²)	b (%/rad)	c (%)	RMS Near Infrared (%)
LAI = 1	79	1.08	0.48	4.96	0.18	14.12	0.50	36.14	1.85
"	40	−0.58	0.51	6.52	0.06	0.90	2.64	30.16	2.21
"	17	−0.89	0.27	6.89	0.05	4.73	1.43	26.95	0.29
LAI = 2	79	1.55	0.49	3.64	0.19	16.05	0.50	43.30	2.10
"	40	0.18	0.63	3.84	0.09	1.54	3.38	37.65	2.95
"	17	−0.16	0.35	4.01	0.05	6.67	1.83	33.01	0.36
LAI = 4	79	1.76	0.49	3.29	0.20	15.49	0.49	52.40	2.16
"	40	0.80	0.68	2.70	0.10	1.92	3.59	49.26	2.76
"	17	0.60	0.38	2.58	0.06	6.23	1.97	44.67	0.40

Surface	Solar Zenith (deg)	Visible Reflectance Coefficients			RMS Visible (%)	Near Infrared Reflectance Coefficients			RMS Near Infrared (%)
Surface	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Visible (%)	a (%/rad²)	b (%/rad)	c (%)	RMS Near Infrared (%)
Soybeans	38	0.66	0.31	3.00	0.22	5.08	5.27	46.22	2.38
LAI=2.58	30	0.48	0.50	2.95	0.15	4.10	5.90	43.43	1.33
"	39	0.72	0.65	2.97	0.16	5.90	6.70	43.96	1.73
"	61	1.49	0.32	3.44	0.25	9.09	7.62	46.76	2.31

Surface	Solar Zenith (deg)	Solar Reflectance Coefficients^a			RMS Solar (%)	RMS Eq. (4) (%)	Hemi spherical Reflectance^b (%)
Surface	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Solar (%)	RMS Eq. (4) (%)	Hemi spherical Reflectance^b (%)

Rough Soil	20	−2.08	3.61	13.65	1.47	1.65	12.22
"	28	−1.88	4.77	14.98	1.45	1.89	13.78
"	63	4.64	10.04	10.33	3.33	3.62	13.46
"	69	5.05	8.86	8.43	3.34	3.40	11.35
Medium Soil	28	−2.48	2.05	13.91	3.08	1.61	12.61
"	43	−0.74	5.61	12.15	1.57	1.76	11.58
Smooth Soil	38	0.18	1.79	7.85	0.64	5.14	8.01
"	44	1.09	2.24	6.88	0.70	5.50	7.70

Surface	Solar Zenith (deg.)	Visible Reflectance R_N/R_H^a	Reflectance R_N/R_H^b	Near Infrared R_N/R_H^a	Reflectance R_N/R_H^b
Simulated	79	0.86	0.92	0.78	0.86
LAI = 1	40	1.07	1.04	0.98	0.92
"	17	1.11	1.06	0.89	0.93
Simulated	79	0.76	0.85	0.79	0.86
LAI = 2	40	0.97	0.98	0.97	0.91
"	17	1.03	1.02	0.87	0.92
Simulated	79	0.72	0.82	0.82	0.88
LAI = 4	40	0.82	0.89	0.97	0.93
"	17	0.86	0.91	0.91	0.94
Soybean	38	0.86	0.94	0.93	0.97
LAI =2.87	30	0.89	0.94	0.94	0.97
"	38	0.85	0.90	0.91	0.95
"	61	0.76	0.83	0.88	0.93

Surface	Solar Zenith (deg.)	Solar R_N/R_H^a (%)	Reflectance^c R_N/R_H^b (%)

Rough Soil	20	1.11	1.10
"	28	1.09	1.07
"	63	0.67	0.68
"	69	0.73	0.82
Medium Soil	28	1.15	1.15
"	43	1.05	1.06
Smooth Soil	38	0.98	0.85
"	44	0.90	0.97

Canopy	Solar Zenith (deg)	Visible Reflectance Coefficients			RMS Visible (%)	Near Infrared Reflectance Coefficients			RMS Near Infrared (%)
Canopy	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Visible (%)	a (%/rad²)	b (%/rad)	c (%)	RMS Near Infrared (%)
LAI = 1	79	1.08	0.48	4.96	0.18	14.12	0.50	36.14	1.85
"	40	−0.58	0.51	6.52	0.06	0.90	2.64	30.16	2.21
"	17	−0.89	0.27	6.89	0.05	4.73	1.43	26.95	0.29
LAI = 2	79	1.55	0.49	3.64	0.19	16.05	0.50	43.30	2.10
"	40	0.18	0.63	3.84	0.09	1.54	3.38	37.65	2.95
"	17	−0.16	0.35	4.01	0.05	6.67	1.83	33.01	0.36
LAI = 4	79	1.76	0.49	3.29	0.20	15.49	0.49	52.40	2.16
"	40	0.80	0.68	2.70	0.10	1.92	3.59	49.26	2.76
"	17	0.60	0.38	2.58	0.06	6.23	1.97	44.67	0.40

Surface	Solar Zenith (deg)	Visible Reflectance Coefficients			RMS Visible (%)	Near Infrared Reflectance Coefficients			RMS Near Infrared (%)
Surface	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Visible (%)	a (%/rad²)	b (%/rad)	c (%)	RMS Near Infrared (%)
Soybeans	38	0.66	0.31	3.00	0.22	5.08	5.27	46.22	2.38
LAI=2.58	30	0.48	0.50	2.95	0.15	4.10	5.90	43.43	1.33
"	39	0.72	0.65	2.97	0.16	5.90	6.70	43.96	1.73
"	61	1.49	0.32	3.44	0.25	9.09	7.62	46.76	2.31

Surface	Solar Zenith (deg)	Solar Reflectance Coefficients^a			RMS Solar (%)	RMS Eq. (4) (%)	Hemi spherical Reflectance^b (%)
Surface	Solar Zenith (deg)	a (%/rad²)	b (%/rad)	c (%)	RMS Solar (%)	RMS Eq. (4) (%)	Hemi spherical Reflectance^b (%)

Rough Soil	20	−2.08	3.61	13.65	1.47	1.65	12.22
"	28	−1.88	4.77	14.98	1.45	1.89	13.78
"	63	4.64	10.04	10.33	3.33	3.62	13.46
"	69	5.05	8.86	8.43	3.34	3.40	11.35
Medium Soil	28	−2.48	2.05	13.91	3.08	1.61	12.61
"	43	−0.74	5.61	12.15	1.57	1.76	11.58
Smooth Soil	38	0.18	1.79	7.85	0.64	5.14	8.01
"	44	1.09	2.24	6.88	0.70	5.50	7.70

Simple equation to approximate the bidirectional reflectance from vegetative canopies and bare soil surfaces

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Applied Optics