## Canopy Reflectance Model Inversion

Canopy reflectance models relate fundamental surface parameters, for example, LAI, leaf optical properties, to scene reflectance for a given sun-surface-sensor geometry. These models may be grouped according to four categories: parametric, geometrical, turbid medium, and computer simulation models. Parametric models are based on simple mathematical functions of reflectance distribution, and the others on physical radiative transfer process in the canopy. These models have been used to estimate canopy morphological and optical properties.

### Canopy reflectance models

Parametric models (sometimes called semiempirical models) represent the realistic canopy reflectance distributions through simple mathematical combinations of various terms or kernels. In general, it takes the form of p = aK0 + bK1 + cK2

where K0, Kb and K2, the kernels, are functions of the view and illumination geometry, and the constants a, b, and c describe the nonangular contributions to the canopy reflectance (and are functions of canopy biophysical parameters). One example of a simple parametric model takes the form of p = a62 + b0cos(^' - 4>) + c

Where a, b, and c are constants which implicitly depend upon the solar and canopy geometry, 0 is the view zenith angle, and ^ are the solar and view azimuth angles, respectively.

Geometric models simulate heterogeneous scenes with geometric objects (e.g., cones, spheroids, ellipsoids, and cubes) protruding from a surface. Optical behavior of objects is constant or simplified. Reflectance anisotropy is determined primarily by the fractions and spatial orientation of shaded and sunlit surfaces, both canopy and ground, for a particular sun-target-sensor geometry. Geometric models have been inverted to retrieve canopy structural and optical information.

Turbid-medium models simulate a canopy with a cloud of infinitesimal platelets having the optical properties of plant organs (most commonly leaves). The models generally include single and multiple scattering; the former is determined quasi-analytically, while the latter must be approximated. The canopy architecture is described at the leaf level through LAI and LAD (leaf angle distribution). Figure 6 shows an example of the simulated reflectances in red and NIR bands, respectively, with different LAI values. These models, best suited for dense canopies with small vegetation elements, have successfully simulated radiative transfer in grasses, agricultural crops, and forests. Many turbid-medium models have been inverted to estimate LAI and other parameters.

View zenith angle View zenith angle

Figure 6 Example of the canopy red and near infrared (NIR) reflectances stimulated with a turbid-medium model at various LAI values (0.5,1.0,1.5, 2.0, 2.5, 3.0, 4.0, and 5.0). The solar zenith angle is 50°. Reproduced from Fang H, LiangS, KuuskA(2003) Retrieving leaf area index using a genetic algorithm with a canopy radiative transfer model. Remote Sensing of Environment 85(3): 257-270, with permission from Elsevier.

Computer simulation models use the positions and orientations of various vegetation elements to describe a canopy. One model type uses Monte Carlo methods which rigorously trace photo interactions with an arrangement of discrete scatterings within the canopy. Although computationally expensive, these models accurately simulate within-canopy spatial heterogeneity (e.g., organ size distribution and leaf clumping, gaps) and scene-scale heterogeneity (e.g., topography) that other models must either neglect or approximate with quasi-empirical formulations. While several models have been developed, their computational expense reduces the likelihood that they will be adopted for inversion applications.

### Model inversion and parameter estimation

To calculate canopy biophysical parameters, that is, LAI, the canopy reflectance model has to be solved. Model inversion has been mainly applied over the directional distribution of reflectance. The general reflectance model inversion problem may be stated as follows: given a set of empirical reflectance measurements, one must determine the set of canopy biophysical variables, so that the computed reflectances best fit the empirical reflectances. The fit of the empirical data is determined by a merit function, e2, defined as n B

where r j is the observed directional reflectance for a given viewing and solar angle geometry, r j the simulation model estimate, n the number of reflectance samples, B the number of spectral bands, and W represents the weight. A penalty function may be used to limit the independent parameter space to physically possible values. The ability to correctly determine target parameter space through model inversion, therefore, depends on the dataset rj, the model's likeness to physical reality, and the chosen-optimization algorithm's ability to minimize e2 over the parameter space. The choice of a particular optimization method depends on the mathematical properties of the function to be minimized.

### Hybrid Inversion

The hybrid inversion method combines the advantages of the empirical regression method and the physical model inversion method in order to estimate surface parameters efficiently. For operational application, two inversion techniques, that is, neural networks (NN) and lookup tables (LUT), are commonly used. These two inversion techniques provide a simple solution to the traditional iterative methods requiring hundreds of runs of the model for each pixel. The two methods differ in the way they solve the inverse problem.

The LUT method replaces the runtime computation with a simpler lookup operation and thus can speed up the inversion process significantly. The LUT makes use of the precomputed model reflectances from a large range of input parameters. The lookup process determines the distribution of the precomputed canopy realizations that minimize a given merit function with respect to the observations. This function usually characterizes the distance between the actual reflectance measurements and the simulated reflectances of the training database, possibly weighted by the uncertainty on measurements and model. This method is easy to use since most of the complications lie in generating the database. The accuracy of the LUT method is dependent on the accuracy of canopy radiative transfer models.

NN provide a very efficient tool to establish the relationship between the simulated reflectance field and the corresponding biophysical variable of interest. The fundamental concept of NN is based on biological neurons. The process of an NN inversion may be outlined as follows: (1) given a set of empirical environmental, leaf, canopy and soil parameters, construct a database of canopy reflectances with a forward radiative transfer model; (2) initiate the NN training (or learning) process with part of the database obtained in the first step, and find the relationship between the input data and the output reflectances; (3) check the NN training with the other part of the database or ground measurements; and (4) apply the trained and checked NN model to a new scenario to predict the input parameters. The database must be general enough to include all the possible variations.