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

Transcription

1 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 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. 1. Introduction The bidirectional reflectance distribution function, a fundamental property describing the reflectance characteristics of a surface, has received increased attention in recent years in remote sensing. The soil or canopy bidirectional reflectance distribution is defined here as the reflectance at a multitude of possible view angles at a given time or solar position (see Nicodemus 1 for rigorous definition). Complex radiative transfer models have been used to identify the essential components of plant canopy and bare soil reflectance. 2 4 Our present understanding of how reflectance may vary with different environmental and physiological factors derives from a combination of modeling and field measurements of reflectance. A major difficulty limiting use of canopy or soil bidirectional information is the large volume of data, i.e., an entire 2-D distribution (zenith and azimuth view angles) for each sun angle for each site. A set of soil or canopy bidirectional reflectance values at a single site at ten solar zenith angles can easily consist of 1000 numbers. The focus of past research has been on cause and effect relationships (i.e., characteristics of the surface J. M. Welles is with LICOR, Inc., Lincoln, Nebraska 68504; G. Campbell is with Washington State University, Department of Agronomy & Soils, Pullman, Washington The other authors are with University of Nebraska, Lincoln, Nebraska 68583; C. L. Walthall and B. L. Blad are in the Center for Agricultural Meterology and Climatology, J. M. Norman is in the Department of Agronomy. Received 29 September /85/ $02.00/ Optical Society of America. structure and environment contributing to the observed patterns). The widespread use of bidirectional reflectance data will not occur until we have a simple equation describing the distribution of reflectances as a function of zenith and azimuth view angles with a few parameters. A simple equation describing reflectance distributions provides several important immediate applications. For example, it could serve as (1) an equation for soil bidirectional reflectance as a lower boundary condition in complex plant canopy models, (2) a simple equation for canopy bidirectional reflectance as a lower boundary condition on atmospheric radiative transfer models, and (3) an additional tool for studying directional reflectance effects on wide angle and/or pointable sensors. Such an equation has proven difficult to formulate because it must accommodate a wide range of conditions such as plant type, plant density, and solar zenith angle change. This paper describes a simple equation that describes the bidirectional reflectance of visible and near-ir radiation as a function of view zenith and azimuth angles for several canopy and bare soil surfaces. 11. Background Prior research has given a reasonably clear idea of the bidirectional reflectance distributions from healthy vegetation and bare soil under clear sky conditions. Efforts at modeling visible and near-ir solar energy interactions with vegetation canopies 2-4 have generally produced results which agree well with measurements taken in SitU. 5, 6 From these studies some general observations about vegetation canopy reflectance can be made: (a) reflectance generally increases with increasing zenith view angle for all azimuthal angles; (b) the greatest increase in reflectance usually occurs in the principal plane due to backscatter in the direction of the 1 February 1985 / Vol. 24, No. 3 / APPLIED OPTICS 383

2 sun; (c) increased solar zenith angles tend to generally increase reflectance. Each of these characteristics has been found to be highly dependent on environmental, physiological, morphological, and optical properties of the canopy and the underlying surface. Considerable research is directed toward identifying and determining the roles of each of these properties. Soil bidirectional reflectance distributions arise mainly from shadowing effects of aggregates. These distributions are similar in shape to canopy distributions and can be even more asymmetric or non-lambertian * Fig. 1. Polar contour plot of vegetative canopy visible reflectance predictions from Cupid for a solar zenith angle of 790 and a leaf area index of 4. The distance from the origin represents the view zenith angle indicated by the dotted grids at 15,30,45, and 600. The angle from 0 represents the view azimuth angle. Solar azimuth is at 0. The isolines represent contours of percent reflectance. Visible leaf reflectance and transmittance used in Cupid were 8 and 7%. Near-IR leaf reflectance and transmittance were 46 and 52%. Soil visible reflectance was 14%, and soil near-ir reflectance was 23%. 4e 111. Equation Development An examination of soil and canopy bidirectional reflectance distributions for a range of crop types, solar zenith angles, or leaf area indices reveals a general similarity of surface shapes. To obtain canopy reflectance distributions that could be used to explore simple forms of equations with few coefficients, eighteen different soybean canopy reflectance distributions were simulated using a deterministic model, Cupid. 7 Spherical leaf angle distributions under clear sky conditions were used in the generation of visible ( ,um) and near-ir ( Mum) reflectances for three levels of leaf area index (LAI) under three different sun angle conditions. This spherical distribution is typical of corn or soybeans and many other canopies. Specified leaf visible reflectance and transmittance were 8 and 7%, respectively. Near-IR leaf reflectance was 46%, and transmittance was 52%. Visible soil reflectance was 15%, and near-ir soil reflectance was 23%. The output from Cupid consisted of estimated reflectances at azimuth angles ranging from 0 to 3600 at 450 increments and view zenith angle positions of 7, 22, 30, 45, and 60 (Fig. 1). The 2-D contours of the canopy reflectance distribution appear to be similar to the shape of the limacon of Pascal. A limacon equation r = a + b cos(o) (1) was modified, and coefficients were calculated via a least-squares fitting procedure to fit the general shape to a single contour. The rms difference between a distribution generated using the limacon equation and the original data is used as an indicator of the fit. This fitting/comparison procedure would have required fitting a separate equation to each contour but was discarded when the procedure resulted in unsatisfactorily high rms values. Using the simple limacon equation as a starting point, other equation forms were used in an attempt to fit the 3-D reflectance surfaces directly. Satisfactory results were obtained with the following equation: Table I. Results of Least-Squares Fitting of Eq. (2) on Simulated Canopy Data Canopy Solar Visible Reflectance RMS Near Infrared Reflectance RS Zenith Coefficients Visible Coefficients Near (deg) a 2 b c (%) a b c Infrared (%/rad ) (W/rad) (%) (%/rad 2 ) (/rad) (%) (%) IAI LAI IAI APPLIED OPTICS / Vol. 24, No. 3 / 1 February 1985

3 r = a02 + b, cos(fo - Oks) + c, (2) where r is the reflectance at a given view zenith Ov and view azimuth Xv look angle; a, b, and c are coefficients derived using a linear least-squares fitting procedure, and qs is the solar azimuth angle. Overall, the rms difference in reflectance between Eq. (2) and the model Cupid ranged from 0.1 to 0.2% for red and 0.3 to 3% for NIR (Table I). The rms differences indicate that the best fit of the equation was at small solar zenith angles for both red and near-ir distributions. The poorest fits are the large solar zenith angle distributions for the red and medium zenith angles for NIR. The potential adaptability of the equation to a wide range of canopies is suggested by the extreme surface shapes that the equation successfully fits. The rms differences are not appreciably different for the different canopy densities. The various terms of Eq. (2) control different surface features. The first term (ao') of Eq. (2) controls the general surface curvature. The second term [b0v cos( 0 v - k 8 )] provides a linear dependence on view zenith, which interacts with the first quadratic term to fit more variable surface shapes, and includes a view azimuth dependence that is symmetric about the solar azimuth (principal plane). This assures that reflectance will decrease away from the azimuth of the sun (as the cosine term approaches -1). The constant, or third term, is the nadir reflectance value. Equation (2) can be integrated over the hemisphere of view angles to provide an analytical expression for the surface hemispherical reflectance in terms of the coefficients a, b, and c. The result is 2.305a RH= +C. (3) It is interesting to note that the second term does not contribute to the hemispherical reflectance. Equation (3) is very useful for converting nadir reflectance measurements from aircraft or satellites into hemispherical reflectance measurements. Such hemispherical reflectance estimates are essential if satellites or aircraft are to be used to estimate surface albedo for use in energy balance studies. I'~~~~~~~~~~~~~~~~~~. l Fig. 2. A perspective 3-D plot of LAI = 2.87 soybean field data for O. = 390 (dashed lines) and 0, = 610 (solid lines). The center of the plot represents nadir with the distance from it representing view zenith in lines of 100 increments. View azimuth is represented by the angle from 0. The reflectance at a given view zenith and azimuth is represented by the height of the surface at that point. The scale is in percent reflectance. The third quadrant has been removed for ease of viewing. Solar azimuth is 00. This plot illustrates the increased reflectance resulting from an increase in solar zenith angle and the increase in reflectance with increased view zenith angle Table II. Results of Least-Squares Fitting of Eq. (2) on Soybean and Soil Field Data Surface Solar Visible Reflectance 1S Near Infrared Reflectance R1S Zenith (deg) Coefficients c Visible Coefficients Near a 2 b (%) a b c Infrared (%/rad ) (%/rad) (%) (%/rad2) (/rad) (%) (%2 Soybeans IAI= Surface Solar Solar Reflectance RS R4S Hemispherical Zenith Coefficientsa Solar Eq.(4) Reflectanceb (%) (x) () (W/rad ) (%/rad) (%) Rough Soil a b 2 Medium Soil 28, " Smooth Soil " a Silicon cell response corrected to solar. b Calculated using r cos () Sin (AV) d6vd4v f cos (V) Sin (Ov) d6vdlv 1 February 1985 / Vol. 24, No. 3 / APPLIED OPTICS 385

4 IV. Tests on Field Data Equation (2) was tested by least-squares fits to soybean field data acquired from the Laboratory for Applications of Remote Sensing, Purdue University 8 (Fig. 2). The rms differences range from 0.2 to 0.3% for red and 1.3 to 2.3% for NIR (Table II). The similarity of these results to the modeled data is surprising considering that the field data also include measurement and sampling errors. Equation (2) was tested further on eight soil bidirectional reflectance data sets of rough, medium, and smooth soil surfaces (Table II). The rough surface was recently plowed from sod with furrows -15 cm deep. The medium surface had 2-5-cm clods from multiple tillages, and the smooth surface was a gravel parking lot with cm gravel. The sensor used for these measurements is a collimated silicon cell that is normalized to solar irradiance by integrating the measured soil bidirectional distribution and setting the result of the integral equal to the hemispherical reflectance measured with an Eppley model 2 pyranometer. The rms differences for the soil data range from 0.6 to 3.3% (Table II). The best fits are obtained on the smooth soil, while rough soil surfaces with large solar zenith and the medium soil surface with small solar zenith angle have the poorest fits. A single simple equation to describe the soil bidirectional reflectance distribution with solar incidence angle would be extremely useful in modeling plant canopy bidirectional reflectance, because the equation could provide the soil boundary condition. The coefficients in Table II for the rough and medium soil surfaces can be plotted against solar zenith angle and fit with linear or quadratic curves. The resulting soil reflectance equation based on medium and rough surface data is rsoi = RH"oil [AO + BO, cos(o, - 0.) + C, A = X X 10-50, X , B = X X 10-30, (4) C = X 1-20, x , where RH,SOi is the soil hemispherical reflectance. The rms difference between Eq. (4) and the measured data is shown in Table II, and an example fit is shown in Fig. 3. Although the fit of Eq. (4) to data (Fig. 3) is good, several features in the data are not represented by the fit, notably some apparent furrow structure and the hot spot which appears at the zenith view angle of 43. Equation (4) does not fit the smooth surface data very well. This suggests that surface roughness may be important in soil bidirectional distributions. The ratio of nadir reflectance to hemispherical reflectance can be obtained from Eq. (3) or calculated directly from the measured reflectance data by performing a numerical integral (Table III). These values compare reasonably with measurements of Salmonson and Marlatt 9 ; their values range from 0.41 to 0.88 for solar zenith angles from 56 to 820 over a dry desert lake bed, a sparse grassland, and a vegetation swamp. In Table Ill. Ratio of Nadir Reflectance RN to Hemispherical Reflectance RN for Simulated Vegetation Canopy Data, Soybean, and Soil Field Data Surface Solar Visible Reflectance Near Infrared Reflectance Zenith RN/Re RN/RHb RN/Rje RN/RHb (deg.) Simulated IAI Simulated IAI Simulated IAI Soybean LAI Surface Solar Solar Reflectancec Zenith RN/Rip RN/RHb (deg.) (5) () Fig. 3. Perspective 3-D plot for the O = 430 medium soil data (solid lines) with an overlay of the Eq. (4) fit surface (dotted lines). The distance from nadir is given in 100 increments. Vertical scale is in percent reflectance. The third quadrant has been removed for ease of viewing. Solar azimuth is 00. This figure illustrates how well the equation fits the bidirectional reflectance distribution surface. Note how the equation surface generally approximates the data surface except for the hot spot reflectance peak (where the view zenith equals the solar zenith). Also note how the equation surface increases with the data surface in the solar azimuth direction. Rough Soil Medium Soil " Smooth Soil " a RH calculated using Eq. (3). b RH calculated using f r I cos (v) cs cos) (0,) Sin (v) dvdfv Sin (v) dvd4v c Silicon cell response corrected to solar. 386 APPLIED OPTICS / Vol. 24, No. 3 / 1 February 1985

5 general, the ratio of nadir to hemispherical reflectance tends to be larger at low zenith angles and smaller at high zenith angles. The ratio can exceed unity. V. Conclusions A simple three-term equation appears adequate for describing the bidirectional reflectance of some vegetative canopies and soil surfaces as a function of zenith and azimuth view angles and solar azimuth angles under clear sky conditions. The equation adjusts to canopies of different vegetation density under different solar angles. It works for both simulated and field measured distributions for red and/or near-ir wavebands. A single equation which can be used as a lower boundary condition on vegetation bidirectional models is obtained for estimating soil bidirectional reflectance as a function of solar zenith and azimuth angle and view zenith and azimuth angles. The ratio of nadir to hemispherical reflectance can be estimated directly from two of the three parameters of the simple equation. Thus, given measurements of canopy bidirectional reflectance from three or more view angles, the bidirectional distribution can be estimated as can the ratio of nadir to hemispherical reflectance. References 1. F. E. Nicodemus, Ed., Self Study Manual on Optical Radiation Measurements: Part 1-Concepts, Chapters 1 to 3, Natl. Bur. Stand. U.S. NBS Tech. Memo (National Bureau of Standards, Gaithersburg, Md., 1976). 2. G. H. Suits, "The Cause of Azimuthal Variations in Directional Reflectance of Vegetation Canopies," Remote Sensing Environ. 2, 175 (1972). 3. J. Smith and K. Ranson, MRS. Bidirectional Reflectance Literature Survey (ORI, Silver Spring, Md., 1979). 4. J. M. Norman and J. M. Welles, "Radiative Transfer in an Array of Canopies," Argon. J. 75, 481 (1983). 5. J. E. Colwell, "Vegetation Canopy Reflectance," in Proceedings, Tenth International Symposium on Remote Sensing of Environment (Environmental Research Institute of Michigan, Ann Arbor, 1974). 6. D. Kimes and J. Smith, "Simulation of Solar Radiation Absorption in Vegetation Canopies," Appl. Opt. 19, 2801 (1980). 7. J. M. Norman, "Modeling the Complete Crop Canopy," in Modification of the Aerial Environment of Plants, B. J. Barfield and J. F. Gerber, Eds. (American Society of Agricultural Engineers, St. Joseph, Mich. 1979), Chap K. J. Ranson, V. C. Vanderbilt, L. L. Biehl, B. F. Robinson, and M. E. Bauer, Soybean Canopy Reflectance as a Function of View and Illumination geometry, Agristars Tech. Report (Laboratory for Applications of Remote Sensing, Purdue U., West Lafayette, Ind., 1982). 9. V. V. Salmonson and W. E. Marlatt, "Airborne Measurements of Reflected Solar Radiation," Remote Sensing Environ. 2, 1 (1971). The authors acknowledge support from the University of Nebraska Agricultural Experiment Station, the NASA Fundamental Research Program (grant NAG5-277), the NASA Graduate Researchers Program (grant NGT ), and the NASA Land Resources Program Multiple-Linear Array Project (contract S19583-D). This is published as paper 7646, Journal Series Nebraska Agricultural Experiment Station. The work reported was conducted under Nebraska Experiment Station Projects 11-33, , and February 1985 / Vol. 24, No. 3 / APPLIED OPTICS 387