Leaf area index estimates using remotely sensed data and BRDF models in a semiarid region

The amount and spatial and temporal dynamics of vegetation are important information in environmental studies and agricultural practices. There has been a great deal of interest in estimating vegetation parameters and their spatial and temporal extent using remotely sensed imagery. There are primarily two approaches to estimating vegetation parameters such as leaf area index (LAI). The first one is associated with computation of spectral vegetation indices (SVI) from radiometric measurements. This approach uses an empirical or modeled LAI-SVI relation between remotely sensed variables such as SVI and biophysical variables such as LAI. The major limitation of this empirical approach is that there is no single LAI-SVI equation (with a set of coefficients) that can be applied to remote-sensing images of different surface types. The second approach involves using bidirectional reflectance distribution function (BRDF) models. It inverts a BRDF model with radiometric measurements to estimate LAI using an optimization procedure. Although this approach has a theoretical basis and is potentially applicable to varying surface types, its primary limitation is the lengthy computation time and difficulty of obtaining the required input parameters by the model. In this study, we present a strategy that combines BRDF models and conventional LAI-SVI approaches to circumvent these limitations. The proposed strategy was implemented in three sequential steps. In the first step, a BRDF model was inverted with a limited number of data points or pixels to produce a training data set consisting of leaf area index and associated pixel values. In the second step, the training data set passed through a quality control procedure to remove outliers from the inversion procedure. In the final step, the training data set was used either to fit an LAI-SVI equation or to train a neural fuzzy system. The best fit equation or the trained fuzzy system was then applied to large-scale remote-sensing imagery to map spatial LAI distribution. This approach was applied to Landsat TM imagery acquired in the semiarid southeast Arizona and AVHRR imagery over the Hapex-Sahel experimental sites near Niamy, Niger. The results were compared with limited ground-based LAI measurements and suggested that the proposed approach produced reasonable estimates of leaf area index over large areas in semiarid regions. This study was not intended to show accuracy improvement of LAI estimation from remotely sensed data. Rather, it provides an alternative that is simple and requires little knowledge of study target and few ground measurements.


INTRODUCTION
vegetation indices (SVI) such as the normalized differ-r is the model output (reflectances, or radiance). Model inversion is a process in which the model is run in a re-ence vegetation index (NDVI): verse mode, that is, the inputs to the inversion procedure NDVIϭ(q NIR Ϫq Red )/(q NIR ϩq Red ), (1) are the reflectances (r) and the output is a set of the pawhere q is spectral reflectance in the red and near-infrarameters. The technique commonly used to invert a model red (NIR) region. A common procedure to estimate LAI is to adjust the model parameters in such a way that the is to establish an empirical relationship between NDVI model-predicted values closely match the measured values and LAI by statistically fitting observed LAI values to the (e.g., Pinty et al., 1990;Goel and Thompson, 1984a-b). corresponding NDVI. Among proposed LAI-SVI rela-Most commonly used models are bidirectional reflectance tions are the following forms (Baret, 1995;Best and Har-distribution function (BRDF) models. A BRDF model usulan, 1985;Curran, 1983;Asrar et al., 1985a,b; Peterson ally consists of a set of equations that relate surface physet al., 1987;Price and Bausch, 1995): ical properties to the observed signals as a function of wavelength. The physical properties may include soil re-LAIϭax 3 ϩbx 2 ϩcxϩd, (2) flectances, canopy architectures and optical properties, LAIϭaϩbx c , (3) geometric configuration of the sensing systems, as well as the illumination sources. These properties are all neces-LAIϭϪ1/2a ln(1Ϫx), (4) sary input parameters required by BRDF models. Some LAIϭf(x), (5) models may require other input parameters such as sinwhere x is either vegetation indices or reflectances degle scattering albedos of individual leaves, leaf inclination rived from remotely sensed data. Coefficients a, b, c, and distribution, and anisotropic properties of both canopy d are empirical parameters and vary with vegetation types. and soil substrates. Given necessary inputs, bidirectional The last equation is a generic function of any form. Given reflectances can be simulated. If bidirectional refleca set of coefficients, the equations can be applied to retances are available, BRDF models can theoretically be motely sensed images to map the spatial LAI distribuinverted to obtain the model parameters, and some vegetions. The advantage of this approach is its simplicity and tation properties can thus be estimated (Goel and Thompease of computation. son, 1984a,b;Goel and Deering, 1985; Goel and Grier, A major limitation of this VI approach, however, is the 1987; Jacquemoud, 1993; Jacquemoud et al., 1995; Qi et diversity of the proposed LAI-SVI equations. These equaal., 1995;Running et al., 1996). tions vary not only in mathematical form (linear, power, A major advantage of the modeling approach is that exponential, etc.), but also in their empirical coefficients, it is a physically based approach and is independent of depending primarily on vegetation type. To operationally vegetation type. It only requires multidirectional meause this VI approach, an LAI-SVI equation must be estabsurements, which are available from many sensors such lished for each vegetation type, which requires substanas AVHRR, VEGETATION, MODIS, and MISR. tial LAI measurements and corresponding remote-sens-There are two major limitations in operational use ing data. Because there is no universal LAI-SVI equation of a modeling approach with BRDF. The first one is reapplicable to diverse vegetation types, it is difficult to use lated to the inversion process of a BRDF model. Some this approach with large-scale remote-sensing images. models may have multiple solutions, given a set of re-Another limitation of this approach is the sensitivity mote-sensing measurements, and the inversion may not of SVI to nonvegetation related factors such as soil backalways converge (Jacquemoud, 1993). This would result ground properties (e.g., Qi et al., 1993), atin unreliable estimates of biophysical variables. The secmospheric conditions (e.g., Kaufman, 1989;Vermote et ond limitation is the computation time involved in a large al., 1990), topography (Holben and Justice, 1980; Justice number of inversion processes, which is a major barrier et al., 1981;Pinter et al., 1987), and bidirectional nature when using large satellite images. of surfaces (Kimes et al., 1985;Deering, 1989;Jackson In summary, both empirical and modeling approaches et al., 1990;Roujean et al., 1992;Burgess and Pairman, have advantages and limitations. The advantages of the 1997). The effects from soil background variations and empirical LAI-SVI approach are that it is simple and atmospheric conditions may be minimized by developing easy to compute. The limitations of this approach are reimproved vegetation indices (Huete, 1988;Clevers, 1989; lated to variable LAI-SVI equations for different biomes, Kaufman and Tanré , 1992;Qi et al., 1994a,b,c). The inand a prior knowledge about the particular biomes is refluence of Sun-surface-sensor geometry needs to be inquired in order to determine the empirical equation cocorporated in vegetation index development and has not efficients. The advantages of the modeling approach are been done yet.
that it is physically based and biome-independent. How-An alternative to empirical relationships is a modelever, it requires substantial computation time for model ing approach based on a set of radiative transfer equainversion, and inversion may not always converge. Theretions or models. It involves inverting a model, rϭf (LAI, fore, none of the approaches is desirable for operational applications. Consequently, the objective of this study is . . .), where LAI is an input parameter to the model and eling approach is retained. The selected BRDF model in this study was the SAIL model by Verhoef (1984) because this model has been tested in many studies for LAI estimation using inversion techniques (e.g., Jacquemoud, 1993;Goel and Thompson, 1984a,b;Goel and Grier, 1987). The input parameters of this model include optical properties of leaves and background soil, canopy structure (leaf angle distribution: LAD), and canopy density (LAI). Given these parameters, the model can predict the reflectances. More discussions on model inversion can be found in other studies (Goel and Thompson 1984a,b;Goel and Grier, 1987;Goel and Deering, 1985;Jacquemoud, 1993;. We used measured soil optical properties and ground multidirectional reflectance data to obtain leaf optical properties by inverting SAIL model. The spherical LAD distribution was determined to be the best among other distributions by examining the simulated reflectances against field measurements. Therefore, spherical LAD was used in this study, but can be treated as a free variable. The output of this step was the inverted LAI values of the randomly selected pixels and their corresponding reflectances and NDVI values. At this time, if any ancillary data such as measured LAD, leaf optical properties, or in-situ LAI measurements are available, they can be used either as inputs to the model for inversion or as output for the subsequent process.
Step 2: Quality Control The parameters inverted in Step 1 are then passed to a quality control procedure, which checks the parameter boundary conditions to detect inversion failures. A failure to develop an approach that has these advantages, but ciroccurs when the optimization algorithm does not find an cumvents the limitations, thus enabling operational use of optimal solution to the model before the boundary is remotely sensed imagery to estimate vegetation densities.
reached. The boundary conditions can be defined by checking inverted LAI against observed values or against preset boundaries (0ϽLAIϽLAI max ), where LAI max is the METHODOLOGY upper limit of LAI values of the study area. When a pixel We designed an approach that combines the two apfails in the inversion process or its inverted LAI reaches proaches in such a way that the above-mentioned limitaits boundaries, it is discarded. For example, if the intions can be circumvented, while taking advantage of verted LAI value is negative or it exceeds the upper easy computation of the LAI-SVI approach and physical boundary, this pixel is then discarded. The purpose of aspects of the modeling approach. A graphic presentation this step was to eliminate outliers to avoid using them in of the proposed approach is illustrated in Figure 1. It the next step. The output of this step is called the trainconsists of three sequential steps: model inversion, qualing data set. ity control, and integration/LAI mapping.

Step 3: Integration/LAI Mapping Step 1: Model Inversion
Two integration/LAI mapping approaches can be used at This is essentially the same as the modeling approach this step. One is to use the training data set (from Step 2) mentioned previously, except that it does not invert evto fit equations of the form LAIϭf(SVI) and another is to ery single pixel or data point when applied to large-scale use a neural fuzzy inference system for LAI estimation. images. Instead, it randomly selects a limited number of LAI-SVI Approach pixels (Nϭ2500 in this study) and uses them to perform Once the data (LAI, SVI, and reflectance) pass through an inversion process. Because the inversion is performed the quality control step, they are used to fit a suite of on a small number of data points, the computation time is greatly reduced, and the theoretical aspect of the mod-LAI-SVI equations proposed in the literature [Eqs. (2), (3), (4), and (5)]. The purpose of this step is to find the NIR, and NDVI. This procedure shortened the computation time substantially compared with the model inver-optimal corresponding coefficients (obtained otherwise with field measurements) of those equations so that they sion technique, although it took longer when compared with the LAI-SVI approach. can be applied to remote-sensing imagery. The technique used in the optimization in this study was to examine a statistical merit function d 2 , defined as DATA DESCRIPTION To test the proposed approaches, data sets from two field measurement campaigns were used to estimate spawhere subscript e stands for estimated LAI using the setial and temporal LAI patterns from satellite images. The lected equation and subscript m stands for modeled LAI first data set was obtained during the MONSOON'90 exvalues from the inversion step. By examining the correperiment (Kustas et al., 1991) conducted at the Walnut sponding d 2 values, the best-fit equation, along with its Gulch Experimental Watershed (31.72ЊN 110.68ЊW) near coefficients, is then used to map LAI distribution. This Tucson, Arizona in the summer of 1990. This watershed way, both the equation and its coefficients can be autowas representative of the brush and grass-covered rangematically obtained (instead of using published or fieldland found in a semiarid environment, and can be genermeasured values). This step takes advantage of the simally said to have only two seasons: dry and wet. Vegetaplicity and easy computation of the LAI-SVI approach tion growth is triggered primarily by high-intensity and circumvents the limitations of computation time of thunderstorms, which normally occur in the wet season the modeling approach. Because only the selected equa-(July-October). The rest of the year is generally dry and tion is used, any outliers that accidentally passed through the vegetation density is low (LAIϽ2.0). the quality control should be removed as well. We used During this campaign, ground-and aircraft-based bithe NDVI [Eq. (1)] for the LAI-SVI fitting because Eqs.
directional measurements (Huete et al., 1992;Qi et al., (2)-(5) were proposed based on the NDVI analysis in 1994b) were made over two sites, one dominated by detheir original studies. However, other vegetation indices, sert grasses and the other by desert shrubs. Ground vegespecially those that were designed to reduce different etation samples were collected near eight meteorological types of noise (Huete, 1988;Kaufman and Tanré , 1992; stations across the watershed. Although the dates of veg- Pinty and Verstraete, 1992;Qi et al., 1994c), can be used.
etation sampling did not coincide with remote-sensing measurements, the ground LAI data represented values Neural Fuzzy Logic Approach of two seasons (dry and wet), which generally reflect the An alternative to the LAI-SVI approach is to use techdynamics of vegetation densities of this region. Landsat niques such as fuzzy inference system or neural network.
TM images, having nearly nadir view angles, were ac-To demonstrate the use of these techniques, we applied quired on 22 April and 7 September 1990. The raw digian adaptive-neural fuzzy inference system to investigate tal images were converted to surface reflectance using the feasibility of such a technique for operational appliground-based reflectance measurements of known tarcations of remote-sensing. The adaptive neural fuzzy ingets proposed by Moran et al. (1996). Together with airference routine was implemented in MATLAB 1 softborne and ground measurements, these satellite images ware (ANFIS: adaptive neural fuzzy inference system).
were applied to the proposed approach outlined in Fig-Use of this routine involved three steps: training the ure 1 to map spatial and temporal LAI values. fuzzy system with a limited number of data points; The second data set included ground-based radiochecking the performance of the system with a separate metric measurements at the Audubon research ranch, data set; and then applying the trained system to remotenear Elgin, Arizona, coincident with LAI measurements. sensing imagery.
The dominant vegetation types were native upland grasses, We used half of the data points that had passed Lehmann's lovegrass, and sacaton grasses. Radiometric through the quality control ( Step 2) to train the ANFIS measurements were made using an 8-band radiometer system and used the other half to check the system perequipped with TM filters, by walking the radiometer formance. The training and checking data sets included along a preset transect. By ratioing the target readings reflectances in the red and NIR spectral region, comto those measured over a reference panel, reflectance puted NDVI, and inverted LAI values. The trained fuzzy factors were computed. The in-situ LAI measurements inference system was applied to remote-sensing images were made with both destructive sampling and Li-Cor's using the built-in procedure EVALFIS, which generated LAI 2000 instrument. The instrument was calibrated the best "guess" of LAI, given the input variables of red, against the data from destructive samples. The third data set was from the Hapex-Sahel experiment in 1991-1992 near Niamey, Niger (Goutorbe et al.,

Adaptive Neural Fuzzy Inference System RESULTS
The trained adaptive fuzzy system consisted of three in-

Equation Selected
puts, each consisting of two generalized bell-shaped membership functions. The ANFIS consisted of eight Using the ground multidirectional measurements and inlinear membership functions. Because of the large numversion procedures, we obtained a set of parameters from ber of training data points, the C-Means subtractive the SAIL model (q kL , s kL , LAI, q S ), which are listed in Tafuzzy inference system was used. This was to avoid placble 1. Parameters q kL and s kL are single leaf (L) reflectance ing equal weight on every data point in case the training and transmittance while q S is the soil reflectance. They are all wavelength (k)-dependent. Since these values are ob-data consist of unevenly distributed clusters. The perfor-  mance of the neural fuzzy system is presented graphi-ues ranged from 0 to 1.7 for the dry season and 0 to 2.8 for the wet season. cally in Figure 3. The neural fuzzy inference system successfully predicted the LAI values.

Comparison with Field Measurements LAI Estimation
To validate the proposed approach, estimated LAI values using the derived equation [Eq.
(2)] and satellite images The LAI maps for 22 April and 7 September 1990 (Fig. 4) were generated with the LAI-SVI equation [Eq.
were compared with field measurements. The comparison is presented in Figures 7 and 8. In Figure 7a, the (2)] derived from TM subpixels in Step 3 (in Fig. 1). The estimated LAI values ranged from 0 to 0.8 on 22 April LAI values estimated from TM images were compared with ground-based LAI values from eight meteorological and from 0 to 3 on 7 September. The dark solid features were either roads or airstrips, and the white areas were stations across the Walnut Gulch Experimental Watershed. Because the ground LAI measurements were made streams where soil water content is generally greater because of accumulated rainfall. The approach produced on different dates, data acquired prior to 26 July were averaged to represent dry season vegetation density, and LAI maps that agreed with the spatial distribution and temporal dynamics of the vegetation in this area. those data after this date were averaged to represent wet season vegetation density. The dry season ground mea-The result of using the neural fuzzy system to map the LAI was demonstrated with the Landsat TM image surements were compared with LAI values derived from 22 April TM images, while the wet season measurements acquired on 7 September 1990 (Fig. 5). The spatial pattern of the LAI map generated with the fuzzy inference were compared with those from 7 September TM images. Vegetation in this area is generally low for the en-system (Fig. 5a) was similar to that generated with the LAI-SVI equation (Fig. 5b). Statistical analysis (also see tire period of dry season and then reaches a maximum standing biomass shortly after the monsoon season, Fig. 3) indicated that there was no significant difference between the LAI maps generated with the neural fuzzy which normally starts in late July. Although the TM-derived LAI values represent "snapshots" of dry and wet and LAI-SVI approaches. This suggests that the fuzzy inference system provided an easy and operationally feasi-season, they agreed well with ground measurements. Because there was no coincident TM and ground LAI data ble technique to apply sophisticated BRDF models to satellite images for LAI estimation (and possible other available at this site, the results may not be conclusive. To further validate the proposed approach, seasonal physical variables as well) over large areas.
The derived equation [Eq.
(2)] was also applied to ground-based radiometric measurements were used to estimate LAI [using derived Eq. (2)] and compared with AVHRR images acquired at the Hapex-Sahel experiment site to test the applicability of the approach to larger in-situ LAI measurements (Fig. 7b) made at the Audubon research ranch in 1997. At this site, paired ground scale remote-sensing imagery. The AVHRR had a spatial resolution of one kilometer. Figures 6a and 6b depict the radiometric and LAI measurements were available to allow a direct comparison. In Figure 7b, estimated LAI dry and wet seasonal LAI patterns, derived from composited AVHRR images (11N-16ЊN, 0ЊE-5ЊE) of May values using the linear LAI-SVI approach by Asrar et al. (1985b) was also included for comparison. The estimated and September 1992. The LAI maps showed the vegetation density gradient from south to north, due to precipi-LAI values with both linear and derived polynomial equations agreed well with the in-situ measurements. tation differences of this region. The estimated LAI val-   Therefore, the proposed approach appears to have pro-from the 1992 AVHRR imagery. However, ground LAI measurements were made in 1993 only. Nevertheless, duced reasonable estimates of LAI within this region.
In Figure 8a, the ground LAI measurements made comparison of the temporal LAI dynamics (Fig. 8b) provided an indirect validation of the approach for Millet at the South (S) site (13.24ЊN, 2.24ЊE) and Central West (CW) site (13.54ЊN, 2.51ЊE) during the Hapex-Sahel vegetation type for this site. Overall, the estimated LAI values were reasonably close to the ground measurements field campaigns are plotted as vertical bars, while the estimated LAI values from AVHRR data are plotted as for all data from the Hapex-Sahel experimental sites.
To compare the results from this approach with solid lines as a function of day of year (DOY). There was good agreement between the estimated and measured those from other approaches such as the linear LAI-SVI by Asrar et al. (1985b), LAI maps were generated using LAI values up to DOY 280. Considering the differences in spatial sampling schemes of ground measurements 7 September 1990 TM images and compared in Figure  9. The spatial patterns were very similar, although the and AVHRR spatial resolution (1.1 km), the results were reasonably satisfactory. The estimated LAI value on absolute LAI values may differ. To further examine the similarity and differences between the two approaches, DOY 280 was underestimated by a factor of 50% (0.4 vs. 0.8) for the south Fallow site. At the Central West Fal-randomly extracted data (10,000 pixels) from these two LAI maps are compared in Figure 10. When LAIϽ1.2, low site, the estimated LAI values appeared to have been overestimated (Fig. 8a) (maximum difference was 0.18).
there was virtually no difference between the two approaches. However, they deviated from the 1:1 line sig-Again, the vegetation sampling schemes and the larger spatial resolution of the AVHRR data might have con-nificantly when LAIϾ1.2. The deviation from linear fit was due to the non-linear response of NDVI with LAI. tributed to these discrepancies.
LAI values were computed for the South Millet site However, in arid and semiarid regions, green leaf area index values are often small and, therefore, either ap-sification maps first and apply this approach for each vegetation class (Running et al., 1996). Global use of this proach is suitable.
approach is therefore limited to areas of similar vegetation type and phenology.

CONCLUDING REMARKS
The proposed approach requires that multiangular (either ground-or space-based) data be collected for the The proposed approach to estimating LAI using remoteuse in the inversion step in Figure 1. Measuring multidisensing images was demonstrated to be reasonably satisrectional reflectance data may be difficult in many cases. factory with the data from these experiments. The results In this case, direct measurements of the required model from the proposed approach, with limited data sets, were parameters may be used instead. similar to the linear empirical approach by Asrar et al. Because of its robust nature and substantially re-(1985b). This approach was promising in that it did not duced computation time, the proposed approach can be require laborious intensive ground LAI measurements. It operationally implemented. Both LAI-SVI and neural relied only on multidirectional remote-sensing measurefuzzy inference system techniques worked well with the ments from either space-based or ground-based sensing systems. Because this approach is independent of vegeta-data set used. The LAI-SVI technique can be easily understood, but interpretation of fuzzy inference system may tion type, it can be applied easily to images acquired over large areas if the area of interest consists of the be difficult, although they both can be used in generating large-scale LAI maps using remote-sensing images. same or similar vegetation types. Therefore, use of a single LAI-SVI equation derived from one study site on ge- There are several comments about the proposed approach. First, no sensitivity analysis was conducted in ographically different areas should be cautious. In this study, the area of interest consisted primarily of desert this study to examine how noise levels inherent within remote-sensing images could have affected the accuracy grass or shrubs and, therefore, may not be suitable for areas of multiple vegetation types. One way to deal with of LAI estimation. In many cases, multidirectional remote-sensing images are compiled or mosaicked from multivegetation type areas is to generate vegetation clas- several satellite overpasses after georeferencing and at-this study. One should include as many types of equations as possible when using this approach. In addition, mospheric corrections. If any of these images contain noise due to atmospheric perturbation, cloud contamina-the spectral vegetation index used in this study was NDVI. As a number of studies indicated, the NDVI is tion, and soil background variation, the model inversion may not perform well. In this case, it is suggested that subject to many external effects, particularly to soil background and atmospheric conditions. It is suggested that these pixels be excluded in the quality control step by setting up a flag. In some cases, the noise level in satel-other indices be tested. The choice of NDVI in this study was made because this index has been used fre-lite images may not be high enough to result in inversion failure, but high enough to result in substantial errors in quently in the past, and LAI-NDVI equations were proposed in numerous studies. Nevertheless, NDVI has estimated LAI values. One way to avoid this possible error is to ensure the data quality by removing these af-been demonstrated to give satisfactory LAI estimates. A third comment is related to bidirectional effects fected pixels or images.
A second comment is related to the LAI-SVI equa-one may find in remote-sensing images. For the inversion process, the bidirectional information is critically im-tion selection procedure used in this approach. Because the inversion is made with randomly selected pixels, the portant because BRDF models rely on this angular information for successful simulation and inversions. These selected equation and its associated coefficients may be different each time the approach is used. This may be bidirectional effects are no longer useful, however, after the inversion procedure is finished in Step 1. They may resolved by selecting a large representative number of pixels (Nϭ2500 in this study) so that there is no signifi-cause estimation errors, because vegetation indices such as NDVI are quite sensitive to view angles and solar po-cant difference between subsequent data sampling. Furthermore, other types of LAI-SVI equations may per-sitions. Therefore, the bidirectional effects may be propagated to influence LAI estimations when using the form better for a particular data set than those listed in Figure 9. Comparison of LAI maps derived using Eq. (4) and linear fit equation (5) and TM image acquired on 7 September 1990.

LAI-SVI approach in
Step 3. To avoid this type of error, improve the LAI estimation accuracy, because they provide additional information to train the fuzzy system. the inversion can theoretically be done with the entire image (therefore, there is no need for statistical regres-Therefore, the neural fuzzy system approach may be preferable to the LAI-SVI approach in Step 3. sion and equation selection). Doing so, however, may not only require unacceptable computation time, but also re-Finally, the accuracy in LAI estimation depends on the accuracy of BRDF models used. In the SAIL model sult in a substantial number of inversion failures. The inversion failure may result from noise in the satellite im-used in this study, the soil background was treated as a Lambertian surface. This assumption may have contrib-ages as mentioned previously or from the lack of considerations of factors, such as topography in BRDF uted to the discrepancies found in this study. Use of other models, such as PROSPECT (Jacquemoud, 1993), models. The topography should be considered in conjunction with the sensor's geometric configuration, but need to be explored. It should also be pointed out that the proposed approach can be used to estimate other has not yet been incorporated into BRDF models. The bidirectional effects may not be a concern, however, with physical parameters such as leaf optical properties and vegetation geometric structure parameters by inverting the neural fuzzy system approach (Step 3b in Fig. 1), because one can include the angular variables (view and physically based models. Again, the neural fuzzy system may have advantages over the equation-based approach, solar zenith and azimuth angles, for example) in the training data set. The inclusion of angular variables may because there are not many established equations linking eters of row planted vegetation canopies using reflectance data for only four views. Remote Sens. Environ. 21:37-51. Goel, N. S., and Thompson, R. L. (1984a), Inversion of vegeta-