Remote Sensing of Environment 87 (2003) 55 – 71 www.elsevier.com/locate/rse
An interpolation procedure for generalizing a look-up table inversion method J.P. Gastellu-Etchegorry *, F. Gascon, P. Este`ve Centre d’Etudes Spatiales de la BIOsphe`re (CESBIO), (CNES/CNRS/UPS), BPi 2801, 18 Av. Edouard Belin, 31401 Toulouse ce´dex 4, France Received 28 December 2001; received in revised form 23 May 2003; accepted 1 June 2003
Abstract The inversion of physically based reflectance models is increasingly efficient for extracting vegetation variables from remote sensing images. It requires a vegetation reflectance model and an inversion method that are accurate and efficient. Usually, the complexity of reflectance models implies to use specific inversion methods (e.g., look-up table and neural network). Unfortunately, these methods are valid only for the view-sun directions for which they are designed. A developed look-up table based inversion method avoids this limitation: it generalizes any look-up table for any view-sun direction, and more generally for any input parameter value. It uses a look-up table made of ci coefficients of any analytical expression h that fits a set of reflectance values simulated by the Discrete Anisotropic Radiative Transfer (DART) model. Interpolation on coefficients ci allows h to give reflectance values for any input parameter value. We settled some options of the inversion method with sensitivity studies: tree covers are simulated with 4-tree scenes, expression h has six coefficients ci and the interpolation is the continuous first derivative interpolation method. Moreover, the robustness of the inversion method was validated. The ability to generalize a look-up table for any view-sun direction was successfully tested with the inversion of SPOT images of Fontainebleau (France) forest. LAI maps proved to be as accurate (i.e., RMSE c 1.3) as those obtained with classical relationships that are calibrated with in situ LAI measurements. Here, the advantage of our inversion method was to avoid this calibration. D 2003 Elsevier Inc. All rights reserved. Keywords: LAI; DART; SPOT images
1. Introduction The study of continental biosphere implies that vegetation variables be known worldwide, continuously and at different scales of time and space. Fortunately, (1) vegetation variables affect more or less directly the spectral reflectance of earth surfaces, (2) vegetation reflectance models can simulate the reflectance of earth surfaces using input parameters that correspond to vegetation variables and view-sun configurations, and (3) remote sensing allows one to measure the reflectance of earth surfaces, continuously, worldwide and at different scales of time and space. In this context, physically based reflectance models (Gastellu-Etchegorry, Demarez, Pinel, & Zagolski, 1996; Goel & Thompson, 2000; Myneni & Ross, 1991; Quin & Liang, 2000) are
* Corresponding author. Tel.: +33-5-61-55-61-30; fax: +33-5-61-5585-00. E-mail address:
[email protected] (J.P. Gastellu-Etchegorry). 0034-4257/03/$ - see front matter D 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0034-4257(03)00146-9
very useful tools because they can relate accurately a large number of vegetation optical and structural variables to satellite measurements (e.g., SPOT, MODIS, MISR, POLDER, Ve´ge´tation). Being designed to model physical processes that explain vegetation reflectance, these models are potentially more robust and accurate than other vegetation reflectance models such as empirical models, including spectral indices. Physically based models range in complexity from simple nonlinear models to complex numerical radiative transfer models (e.g., Monte Carlo, ray tracing) in realistic threedimensional computer simulations of vegetation canopies. Usually, they require a large number of input parameters because they need to address the entire radiative transfer problem in order to simulate accurate vegetation reflectance values for any experimental conditions (e.g., sun and view directions) and any type of Earth surfaces (e.g., forests, savannas, etc.). For most studies, it is often assumed that a large number of variables are more or less well known (e.g., tree architecture variables depend on the type of landscape
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studied), whereas other variables (e.g., leaf biomass), potentially interesting for environmental studies, are unknown because they vary with time and space. The inversion of canopy reflectance models is potentially a more robust and accurate method than the use of empirical relationships for extracting biophysical parameters from images acquired from space. Indeed, empirical relationships are specific to the image acquisition configuration (e.g., view-sun directions, type of vegetation, sensor spatial resolution, etc.). Goel et al. (Goel & Strebel 1983; Goel & Thompson, 1984; Goel, 1989) were pioneers in the inversion of canopy reflectance models. Actually, the accuracy of the biophysical parameters that are retrieved from reflectance measurements depends a lot on the accuracy of the reflectance model and on the inversion method that are used. This stresses the interest of 3-D physically based canopy reflectance models, because these models are potentially the most accurate ones. A major problem is that the inversion of 3-D reflectance models leads to high levels of complexity (e.g., large number of variables and physical processes, and complex mathematical formulations), a significant increase in required computer resources, a higher potential of ill-posed problems, and many method-specific problems such as sensitivity to noise and initial guesses at the solution. Satellite measurements being more and more available on a regular basis, the scientific community increases its efforts to provide operational inversion algorithms. These efforts have exposed a need to significantly improve the efficiency and accuracy of methods for inverting 3-D reflectance models. The work presented here was completed with this objective in mind. Although many inversion methods are presently available, they are more or less well adapted depending on the problems. Kimes, Knyazikhin, Privette, Abuelgasim, and Gao (2000) divide them into three major categories: (1) Traditional inversion methods (Bacour, 2001; Goel, 1988; Jacquemoud, Bacour, Poilve´, & Frangi, 2000; Pinty, Verstraete, & Dickinson, 1989; Privette, Emery, & Schimel, 1996; Twomey, 1977). They include four main components: (i) measured reflectance data, (ii) a vegetation reflectance model that simulates on the fly vegetation reflectance, (iii) an iterative standard optimization algorithm, and (iv) a figure-of-merit function e2 that must be minimized (Goel & Strebel, 1983). The merit function provides a numerical ‘‘resemblance’’ between measured and simulated reflectance values. When dealing with large sets of satellite imagery, traditional inversion methods cannot be used operationally on a per pixel basis because their computational times are too large. Most used optimization algorithms are the downhill simplex method (Nelder & Mead, 1965), the conjugate direction set method and the quasi Newton method (Brodlie, 1977). (2) Table look-up methods (Este`ve, 1998; Gobron, Pinty, & Verstraete, 1997; Kimes, Gastellu-Etchegorry, & Este`ve, 2002; Knyazikhin, Martonchik, Myneni, Diner, & Running,
1998). Usually, they consist in the search of a pre-computed look-up table of reflectance values in order to find out the reflectance that most resembles the measured reflectance. They can be fast because the most computationally expensive part of the inversion procedure is completed before the inversion itself. Thus, they are well suited to computationally expensive complex 3-D reflectance models. In order to obtain accurate inversion products, reflectance values of the look-up table must be simulated for a wide range of the model-input parameters that most affect landscape reflectance. (3) Neural network methods (Abuelgasim, Gopal, & Stralher, 1998; Kimes et al., 2000, 2002). With the help of a learning set of reflectance measurements/simulations, they are designed to determine the optimal underlying relationship that maps input parameters (i.e., canopy optical and structural parameters) of the reflectance model to outputs (i.e., simulated/measured reflectances). Multi-layer feedforward neural networks have been an influential development in the field of neural networks during the past decade. They can approximate any continuous input – output relation of interest to any degree of accuracy, provided sufficiently many hidden units are used (Cybenko, 1989; Hornik, Stinchcombe, & White, 1989). Traditional inversion methods are potentially robust because they are inherently designed to handle any arbitrary subset of view-sun angles. However, they are computationally intensive because they reach convergence towards the optimal set of free parameters with on the fly simulations. Thus, they are not appropriate for dealing with large sets of remote sensing data on an operational basis. Look-up table and neural network methods are potentially more efficient and accurate. Indeed, (1) they can use complex reflectance models with acceptable computational times, (2) they do not need to decrease the number of input parameters used to characterize earth surfaces or to simplify the physical processes that give rise to surface reflectance, and (3) they do not require initial guesses to model parameters as do traditional inversion methods. Their main disadvantage is to be designed to handle arbitrary subsets of view-sun angles: they cannot be easily generalized to any view-sun angle. Inversion with look-up tables implies to assess scene reflectance for any possible value of the reflectance model input parameters. Let us consider look-up tables of reflectance values simulated for a set of LAIi values. Then, if LAI is between LAIi and LAIi + 1, reflectance can be interpolated to assess scene reflectance. However, this interpolation is not possible with any model input parameter; e.g., it is not possible with sun zenith angle hs because of the hotspot phenomenon. Fig. 1 shows that an interpolation of scene Bidirectional Reflectance Factor (BRFs) simulated with hs = 30j and 50j cannot give the BRF for hs = 40j: it gives unexpected reflectance peaks for view zenith angles hv = 50j and 30j, whereas it does not give the expected hotspot peak at hv = 40j. Thus, a
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2. The DART 3-D canopy reflectance model
Fig. 1. Simulated and interpolated BRFs. Scene BRF with hs = 40j is interpolated over scene BRFs simulated with hs = 30j and 50j.
scene BRF with a Xs sun direction cannot be computed from scene BRFs that are simulated with sun directions Xs1 and Xs2 that bound direction Xs. Interpolation on view angles is also a difficult task, although to a lesser degree. This paper presents an approach that generalizes a lookup based inversion method for any view-sun configuration. First, the DART reflectance model used here is briefly presented. Similarly to other 3-D models, it uses specific landscape simulations. Their impact on the inversion method is discussed in Section 3. The fourth section presents the look-up inversion method and how the generalization for any view-sun configuration is achieved. The fifth section presents major practical steps of the inversion method and some tests of robustness. Finally, the fifth section illustrates the generalization of a look-up table for any view-sun direction with the inversion of SPOT images of the Fontainebleau forest (France).
The Discrete Anisotropic Radiative Transfer (DART) model was designed to simulate radiative transfer in 3D landscapes (e.g., trees, roads, grass, soil) with the exact kernel and discrete ordinate approaches. It works with a discrete simulation of landscapes. It results that any landscape is represented as a rectangular 3-D matrix of parallelepipedic cells of variable dimension containing materials (e.g., leaf, wood, soil, water, etc.). The original DART model was operated with cells that could be only opaque (i.e., horizontal ground) or filled with turbid matter (i.e., leaves, grass, etc.). Hotspot, leaf specular and polarization mechanisms are modeled. Any model simulation is conducted for a specified illumination and sensor responses (i.e., images and BRFs) for a pre-defined set of view directions. DART model was successfully tested against reflectance measurements (Gastellu-Etchegorry et al., 1999) and 3-D models (Pinty et al., 2001). It was used in several scientific works; e.g., study of the impact of canopy structure on satellite images texture (Pinel & Gastellu-Etchegorry, 1998) and determination of the 3D photosynthesis rate and primary production rate of vegetation canopies (Guillevic & Gastellu-Etchegorry, 1999). Moreover, CNES (French Space center) used DART for selecting the spectral domain of the future high spatial resolution PLEIADES-HR satellite sensor (Gascon, Gastellu-Etchegorry, & Lefe`vre, 2001). DART model was recently improved (Gascon, 2001; Gastellu-Etchegorry & Gascon, in preparation). Presently, it simulates radiative transfer simultaneously in the atmo-
Fig. 2. Scene simulation used as an input to DART model. It shows a mixed ‘‘built-up/natural Earth landscape + atmosphere’’ (left) and some examples of DART simulation.
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sphere and earth landscapes. Images can be simulated for different altitudes from the bottom to the top of the atmosphere. Moreover, it works with topography and urban features (i.e., houses, buildings, roads, etc.) that are simulated with plane elements (i.e., triangles and parallelograms). Fig. 2 illustrates the new scene simulation used as an input to DART: pre-defined atmosphere with variable gas, aerosols and water contents over a built-up/natural earth landscape with some topography. It shows also two images simulated for two view directions. Other DART improvements were realized. (1) Simulation for any spectral band in the optical domain, including the thermal infrared domain. (2) Multispectral simulation for decreasing the computational cost of several monospectral simulations. (3) Use of data bases that characterize the atmosphere and various earth materials from 0.3 up to 15 Am. (4) User friendly interface. The mixed spectral regime around 3 Am is simulated with the simultaneous tracking of sun radiation and radiation emitted by the Earth and the atmosphere.
3. Selection of scene simulations for look-up table computation 3.1. Scene simulations for DART simulations The accuracy of inversion procedures strongly depends on the scene simulations (i.e., computer representations) of the earth landscapes that are used by the reflectance model. Scene simulations imply many compromises. Ideally, approximations and simplifications should be kept at a minimum because they tend to decrease the accuracy of simulated reflectance values. On the other hand, simplifications are necessary in order to obtain a scene simulation that is compatible with the reflectance model at hand. Moreover, even in the ideal case of a reflectance model that could work with exact scene simulations, simplifications are necessary to limit computational constraints (i.e., time and volume). With a 3-D model such as DART, the computation of a look-up table of reflectance values can lead to unacceptable computational times due to the very large number of
simulations to perform. Ideally, DART simulations should be conducted with simulations of wide landscapes in order to calculate vegetation reflectance values with the best accuracy. This leads to unacceptable time and volume computational constraints. Thus, we must determine the smaller scene simulation that allows one to operate DART with an acceptable accuracy level. Scenes must be all the more larger since the associate landscapes are heterogeneous. For example, in the case of landscapes with trees, the number of trees in the simulated scene must be large enough to get a good accuracy and small enough to minimize computational time. We considered three types of scene simulations for computing look-up tables. (1) Grass scenes Each grass scene is characterized by a type of soil among nsol pre-defined soils, a discrete grass cover value, a grass species among nf pre-defined species, a grass geometry (mean height and standard deviation) and a grass LAI value among nlu LAI discrete values. A grass species is defined by its leaf optical properties and its leaf angle distribution (LAD). Grass scenes are very simple simulations of natural landscapes: a column of n turbid cells represents an infinite horizontal multi-layer canopy. Consequently, DART simulations are very fast. Unfortunately, the simulation of earth landscapes as grass scenes tends to lead to very inaccurate canopy reflectance values (Gastellu-Etchegorry et al., 1999). This explains why scene simulations of natural landscapes with trees must include trees. (2) 1-Tree scenes 1-Tree scenes are made of a single tree over a square background. Tree ground cover st is simply modified with the variation of the area of the square background. In that case, the actual scene the DART model works with is an infinite scene with a repetitive pattern that is the single tree scene (Fig. 3). 1-Tree scenes used to compute look-up tables are characterized by the type of soil among the nsol predefined soils, the type of understory among the nf predefined leaf species, the understory mean height and standard deviation, the understory LAI among nlh discrete values, the type of leaf species among the nf pre-defined
Fig. 3. Tree scenes (small squares) and associated infinite scenes (large squares) with variable tree ground covers st, from 100% down to 27%.
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Fig. 4. 4-Tree scenes (small squares) and associated infinite scenes (large squares) actually used by DART, with tree ground covers st from 100% down to 32%.
Fig. 5. General structure of the look-up (top) and inversion (bottom) procedures.
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Fig. 6. Arborescent architecture for storing DART files (ns direction files, ns.nf.nc phase functions and ne.nm.nlu.nlt.nt.nc scenes and simulations).
species, the tree LAI among nla discrete values, the trunk geometry (height, diameter, length within tree crown), the tree crown shape (e.g., ellipsoidal, conical, etc.) and dimensions, and finally the tree ground cover value among nt discrete values. The major problem with 1-tree scenes is that even with a very oblique view direction, we can observe unexpected infinite grass strips if tree ground cover is small (i.e., if trees
are no more contiguous). Then, canopy reflectance tends to reach extreme values for view directions that coincide with the directions of the grass strips. Although slightly observed in very regular tree plantations, this effect does not appear in natural vegetation. (3) 4-Tree scenes These scenes are made of four trees over a square surface (Fig. 4), the area of which can be varied to modify tree
Fig. 7. DART directional reflectance values of 1- and 4-tree scenes, for 2 tree ground covers (100%, 75%) and two understory reflectance values (8%, 20%). k = 550 nm. The horizontal axis is for zenith h and azimuth / view angles, with h between 0j and 90j and / between 0j and 360j.
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Fig. 8. DART directional reflectance values of 1- and 4-tree scenes. (a) Red (k = 680 nm) and (b) near infrared (k = 800 nm) spectral domains.
ground cover st. Three trees out of four are located such that canopy reflectance is very accurate for any view-illumination direction. Input parameters of the 4-tree scenes used to compute look-up tables are listed below:
–
– – number nc of spectral bands. – number ne of illumination conditions (i.e., sun direction, proportion of diffuse sky radiation and the N directions that sample the 4p space of directions). – number nf of leaf species (tree crown and understory). Thus, for each leaf species and for each spectral band, we input the top and bottom leaf hemispherical reflectance values, the leaf transmittance, the top and bottom leaf
–
–
roughness, the leaf refraction index n, the leaf angle distribution and the leaf dimension. number ntrunk of trunk types. Thus, for each trunk type and for each spectral band, we input the lambertian reflectance qt and the diameter /. number nsol of soil types. Thus, for each soil, we input the name of the reflectance model (e.g., lambertian or Hapke) and associated coefficients for each spectral band. A digital elevation model can be used. geometry of the understory (mean height, standard deviation), of the trees (height, tree crown shape, etc.) and of the scene (dimensions of the individual cells of the scene). number nlu of understory LAI values.
Fig. 9. DART directional reflectance values of 1- and 4-tree scenes with a 45% tree ground cover; 8% (a) and 20% (b) understory reflectances. k = 550 nm.
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Fig. 10. DART directional reflectance values of 1- and 4-tree scenes with a very important understory reflectance (50%). Differences are strong with a 45% tree ground cover (a). They are negligible with large tree ground covers (b). k = 680 nm.
– number nlt of tree canopy LAI, with a possibility to use leaf density uf instead of LAI. Indeed, uf is a tree characteristic whereas LAI depends on uf and tree ground cover st. – number nt of tree ground cover values st. Fig. 5 displays the general structure of the program that computes look-up tables. It shows also the arborescent architecture used for storing DART files (ns direction files, ns.nf.nc phase functions and ne.nm.nlu.nlt.nt.nc scenes and simulations) (Fig. 6). 3.2. Comparison of the BRFs of 1-tree and 4-tree scenes Figs. 7– 10 display BRFs of 1-tree and 4-tree scenes in the green (550 nm), red (600 nm) and near infrared (800 nm) domains, with 3 tree ground covers st (45%, 75%, 100%) and a 60j sun zenith angle hs. Reflectance is always maximal at the hotspot. DART input parameters such as leaf optical properties (Table 1) and tree dimensions (e.g., crowns with a 15 m height and a 10.5 m diameter) are specific to an oak stand of Fontainebleau forest (Dufreˆne et al., 1997). Here, understory is assumed to be lambertian (qsol = cst).
As expected, reflectance values of 1-tree and 4-tree scenes are very close if tree cover st is large (Fig. 7). Reflectance differences depend on the spectral domain (Fig. 8). Actually, several factors influence reflectance differences. For example, understory plays an increasing role if its reflectance increases and if tree ground cover decreases (Fig. 9). This is especially true for the view directions that coincide with the directions of the strips of understory. For these directions (e.g., /v = 0j and 180j), scene reflectance has local maxima that are all the more pronounced since tree ground cover st is small and if the understory reflectance qsol is large compared to leaf optical properties. This is well illustrated in Fig. 10: the large understory reflectance qsol (50%) and the small tree ground cover st (45%) explain that BRFs of 1-tree scenes have several very strong local maxima, conversely to the smoother 4-tree scene BRFs. Thus, 4-tree scenes are much better adapted than 1-tree scenes for simulating BRFs of landscapes with trees. Consequently, we will use 4-tree scenes.
4. Generalizing a look-up table for any sun-view direction 4.1. Determination of an analytical reflectance model
Table 1 Leaf optical properties of an oak stand of the Fontainebleau forest (Demarez, 1997) Spectral band
Leaf adaxial reflectance (%)
Leaf abaxial reflectance (%)
Leaf transmittance (%)
550 nm (green) 600 nm (red) 800 nm (NIR)
17 10 45
25 6 45
20 5 45
In order to be valid for any input parameter, sun direction Xs and view direction Xv included, the inversion method relies on an analytical BRF model hu,p(Xv). This is parameterized with coefficients ci that depend on a set of free p and fixed u parameters of DART. Coefficients ci are computed such that any hu,p(Xv) fits the DART BRF that corresponds to the set of parameters ( p, u). Consequently, the number of sets of coefficients ci is equal to the number
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of configurations (hs, LAI, etc.) used to conduct DART simulations. Here, the look-up table used in the inversion procedure is made of the coefficients ci. Our inversion method is designed to work with any analytical model hu,p(Xv) such as Kernel Driven Bi-directional Model (KDBM) reflectance models (Roujean, Leroy, & Deschamps, 1992; Wanner, Li, & Strahler, 1995). Here we use a relatively simple analytical model with no azimuth asymmetry because we apply the inversion method to a look-up table that is computed with 4-tree scenes, which implies that there is no azimuth asymmetry. This case is well adapted to natural landscapes such the forest study area considered hereafter. The analytical model (6p model) has six ci coefficients: Rðhv ; Dusv Þ ¼ ðc1 þ c2 hv cosðDusv Þ þ c3 h2v cos2 ðDusv Þ þ c4 h2v sin2 ðDusv ÞÞ 1 ! 1 w 1 þ c5 1 þ tan sv c6 2 Dusv ¼ us uv ; cosðwsv Þ ¼ Xs :Xv The term 1 ! 1 w 1 þ c5 1 þ tan sv c6 2
is for the hotspot, c5 being its height and c6 its width. R has the usual BRF bowl shape with a symmetry to the incidence plane as for most BRFs. If we deal with landscapes the BRF of which has not this symmetry, it should be preferred to use analytical models such as the KDBM models mentioned above. With the assumption that the 6p model may not be accurate enough in some cases, an 8p model was designed as the 6p model with the addition of terms of degrees 4 and 6:
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(a) Least square minimization for computing the c1 to c4 in order to minimize: 0 e2bowl ¼
nX bdirB
B B @ i¼1
wi qs;i 1 1 w 1 þ c5 1 þ tan sv c6 2
wi ðc1 þ c2 hv;i cosðDusv;i Þ þ c3 h2v;i cos2 ðDusv;i Þ 12 C þ c4 h2v;i sin2 ðDusv;i ÞÞA where nbdir is the number of DART discrete view directions and qs,i is DART reflectance for direction i. Weight wi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 100 hv;i for direction i, with hv,i in degrees, allows one to favor directions mostly used in remote sensing. Similarly, the division of qs,i by the term associated with the hotspot allows one to decrease the impact of directions close to the hotspot. Gauss-Jordan elimination method with full pivoting 2 is used to solve the linear system (Bebowl )/(Bcj)=0 b ja {1,2,3,4}, which gives coefficients c1 to c4. (b) Computation of term c5 as a solution of: qs;k ¼ ðc1 þ c2 hv;k cosðDusv;k Þ þ c3 h2v;k cos2 ðDusv;k Þ þ c4 h2v;k sin2 ðDusv;k ÞÞ 1 ! 1 w 1 þ c5 1 þ tan sv c6 2 where k is the index of the direction corresponding to the hotspot. For that direction, we have wsv,k = 0. Consequently, the previous equation becomes: qs;k ¼ ðc1 þ c2 hv;k cosðDusv;k Þ þ c3 h2v;k cos2 ðDusv;k Þ þ c4 h2v;k sin2 ðDusv;k ÞÞ ð1 þ c5 Þ which implies
Rðhv ; Dusv Þ ¼ ðc1 þ c2 hv cosðDusv Þ þ c3 h2v cos2 ðDusv Þ þ c4 h2v sin2 ðDusv Þ þ c5 h4v sin4 ðDusv Þ þ c6 h6v sin6 ðDusv ÞÞ 1 ! 1 w 1 þ c7 1 þ tan sv c8 2
c5 ¼
qs;k c1 þ c2 hv;k cosðDusv;k Þ þ c3 h2v;k cos2 ðDusv;k Þ þ c4 h2v;k sin2 ðDusv;k Þ
1
(c) Computation of term c6 with the minimization of: e2 ¼
nbdir X
qs;i ðc1 þ c2 hv;i cosðDusv;i Þ þ c3 h2v;i cos2 ðDusv;i Þ
i¼1
Coefficients ci of the look-up table are computed with the least square minimization method, independently of the number of coefficients (i.e., 6 or 8). This method minimizes the difference between the BRFs simulated by DART and by the 6p and 8p models, for all parameter values used to create the look-up table: spectral bands, sun direction angles and free parameters. Steps used in order to determine the 6p model are listed below: (1) Initialization of c5 to 0 and c6 to 1. (2) Loop:
þ c4 h2v;i sin2 ðDusv;i ÞÞ 1 !!2 1 w 1 þ c5 1 þ tan sv c6 2 Minimization is performed with the Brent method (Brent, 1973). This is a step by step method that uses, whenever possible, the inverse parabolic interpolation (i.e., e2 is assumed to be parabolic next to minimum) for assessing the value of c6 that minimizes e2.
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Fig. 11. BRFs from DART and the 6p and 8p models. (a) st = 45%. (b) st = 75%. (c) st = 100%. Green (550 nm). qsol = 8%.
Fig. 12. BRFs from DART and the 6p and 8p models. (a) st = 45%. (b) st = 75%. (c) st = 100%. Green (550 nm). qsol = 20%.
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Fig. 13. BRFs from DART and the 6p and 8p models. (a) Red (680 nm). (b) Near infrared (800 nm).
(3) End of the loop if variations of c5, c6 and e2 are smaller than a threshold value. 4.2. Selection of the most appropriate analytical reflectance model Figs. 11– 14 display DART BRFs simulated with 4-tree scenes. They show also the corresponding BRFs simulated by the 6p and 8p models and their root mean square errors (RMSE). The 6p and 8p models fit well most DART BRFs (i.e., low RMSE). Figs. 11 and 12 are for the green spectral domain, for scenes with tree cover equal to 45%, 75% and 100%, and with understory reflectance equal to 8% and 20%, respectively. Reflectance differences between 6p and 8p models on the one
hand, and DART on the other hand are always small with large tree ground covers (e.g., st = 75% or larger). They increase with the decrease of tree cover, especially if understory reflectance is large. As expected, the 8p model is always more accurate than the 6p model, although differences are usually very small. The 6p and 8p models give also good results in the red and near infrared spectral domains (Fig. 13). For very uncommon configurations, the 6p and 8p models cannot accurately approximate DART BRFs. For example, this occurs for DART BRFs simulated with 1-tree scenes that have a small tree cover (e.g., 45%) and an understory reflectance (50%) much larger than tree crown reflectance (Fig. 14). Fortunately, natural landscapes do not display this chaotic BRF behavior.
Fig. 14. BRFs simulated with DART (1-tree scene) and with the 6p and 8p models. (a) st = 45% and qsol = 50%. (b) Same as (a) with st = 75%.
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Fig. 15. Comparison of the three interpolations. Whenever they are fixed, parameter values are: qsol = 0.08, uf = 0.4 m 1, st = 0.75, hs = 20j, hv = 20j, D/sv =180j. (a) Soil reflectance. (b) Leaf density. (c) Tree ground cover. (d) Sun zenith angle.
Two aspects must be considered for selecting between the 6p and 8p models: computational time and model accuracy compared to DART. Simulations showed that both models are accurate, at least for natural landscapes. The 6p model being significantly faster than the 8p model ( c 30% in relative), we selected the 6p model. 4.3. Selection of an interpolation procedure As already mentioned, the inversion procedure uses coefficients ci that correspond to any set of DART input parameter values. These coefficients are computed with an n dimension interpolation on the pre-computed coefficients ci of the look-up table, where n is the number of DART input parameters. Sun angles belong to these parameters. Here, we studied the performance of three one-dimensional interpolations: global polynomial interpolation, cubic spline inter-
polation, local polynomial interpolation with pre-defined first derivatives. The global polynomial interpolation uses the Neville algorithm (Press, Teukolsky, Vetterling, & Flannery, 1992) to assess reflectance values for any parameter value. The cubic spline interpolation uses a specific polynomial between each pair of discrete parameter values. Coefficients of these polynomials are computed such that the interpolated function is continuous through the second derivatives. Local polynomial interpolation uses a specific polynomial between each pair of discrete parameter values, with continuous first derivatives that take pre-defined values for the points of the look-up table. It will be called the ‘‘Continuous 1st derivative’’. More detailed information about these interpolations is given in Este`ve (1998). The way these one-dimensional interpolations are used is illustrated below, with n = 3.
Fig. 16. Comparison of the three interpolations. Whenever they are fixed, parameter values are: qsol = 0.15, uf = 0.6 m 1, st = 0.6, hs = 30j, hv = 20j, D/sv = 180j. (a) Soil reflectance. (b) Leaf density. (c) Tree ground cover. (d) Sun zenith angle.
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Let f( p1, p2, p3) be a function of three parameters p1, p2 and p3. Each parameter is sampled with n1, n2 and n3 discrete values, respectively. Let us assume that x1, x2 and x3 are the parameter values for which we want to assess the value of the function. First step: assessment of f( p1, p2, p3) for each discrete value of p1 and p2 with an interpolation on parameter p3, at x3. This gives n1 n2 values. Second step: assessment of f( p1, p2, p3) for each discrete value of p1 with an interpolation on parameter p2, at x2. This gives n1 values. Third step: assessment of f( p1, p2, p3) with an interpolation on parameter p1, at x1. Analyses were conducted in order to determine the best 1-D interpolation. Only results obtained with the 6p model in the green spectral band are shown here. In a first step, we computed a look-up table of coefficients ci from 4-tree scene BRFs simulated by DART for three sun zenith angles (20j, 40j, 60j), five soil reflectance values qsol (0%, 2%, 32%, 50%, 80%), five leaf density values uf (0.1, 0.2, 0.4, 0.8, 1.6), 4-tree ground covers st (0%, 45%, 75%, 100%) and three spectral bands (green, red, near infrared). Figs. 15 and 16 show the reflectance values that are predicted by the 6p model combined with the three interpolation methods. As expected, the 6p model gives exact values if input parameters are those used to compute the look-up table (e.g., uf = 0.8 and 1.6 m 1). Logically, the global polynomial interpolation gives reflectance values that oscillate around exact DART reflectance values if input parameters are far from those used to compute the look-up table (cf. scene reflectance as a function of leaf density, especially for uf between 0.9 and 1.6 m 1). Thus, we rejected the global polynomial interpolation. Generally speaking, the continuous first derivative interpolation is as accurate as the cubic spline interpolation and it is significantly faster ( c 30% in relative). Consequently, we selected the continuous first derivative interpolation.
5. The inversion procedure Major steps, with some practical aspects, of the inversion method are presented below. 5.1. Selection of free parameters First, the free parameters (i.e., soil reflectance, understory LAI, tree LAI, tree ground cover and leaf chlorophyll concentration) of the inversion procedure must be selected. 5.2. Selection of the soil reflectance model Spectral soil reflectance cannot be a free parameter because the number of free parameters would be too large
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compared to the dimension of the information to invert. For that, we use a rather P general soil parametric model with n parameters: Rs ðkj Þ ¼ ni¼1 pi Rsi ðkj Þ, where kj is for the different spectral bands and n is between 1 and 3. It is coherent with the fact that factors such as humidity tend to have a linear influence over the whole spectrum. Values Rsi(kj) are given by the operator. Consequently, soil reflectance is determined through the calculation of n parameters pi. 5.3. Inversion The inversion is conducted with images acquired in different spectral bands, possibly with different view directions. It can be applied to portions of images if a mask file is available. Generally speaking, the operator must specify: – – – – – – –
the type and name of data to invert (BRF, satellite image). the type of analytical reflectance model (6p or 8p model). the name of the mask file, if necessary. the type of error to minimize (absolute or relative). the type of interpolation. values of fixed parameters. the type of soil model (number of parameters).
The inversion procedure is designed to minimize one among three possible merit functions: – absolute error: e2 ¼
nbval X
ðqi Rðhsi ; hvi ; Dusvi ; ki ; pÞÞ2 ;
i¼1
where nbval is the number of available reflectance values, qi is the ith reflectance value, hsi, hvi and Dusvi describe the sun and view directions, ki is the wavelength of the reflectance value, and p is the free parameters vector. – relative error: nX bval qi Rðhsi ; hvi ; Dusvi ; ki ; pÞ 2 e2 ¼ : qi i¼1 This merit function is designed such that all spectral bands tend to have the same impact on e2. – relative error with threshold: nX bval qi Rðhsi ; hvi ; Dusvi ; ki ; pÞ 2 e2 ¼ ki i¼1 with ki = qi if qi z qmin or otherwise ki = qmin, where qmin is specified by the operator. This threshold qmin is introduced in order not to give too much weight to reflectance values close to 0. The minimization algorithm is a modified simplex method (Nelder & Mead, 1965). Compared to traditional techniques,
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this nonlinear multi-dimensional optimization technique is robust, finds the optimum solution more reliably and requires less a priori information and weaker assumptions. A simplex is a geometric figure with N + 1 vertices in N dimensions. The method starts with an initial simplex of N + 1 points that expands and contracts to adapt to the functional surface and attempts to surround the optimum point. This is achieved with a series of geometric transformations where the current worst point is discarded and replaced by a better one. Once one simplex has terminated, the procedure is repeated with a new simplex. The process is repeated until the simplex collapses onto the same solution, which increases the likelihood of finding the global optimum. 5.4. Constraints on free parameters Because interpolations are conducted on parameter values, any free parameter value must be bounded by parameter values (e.g., soil reflectance is between 0 and 1) that were used to compute the look-up table. Consequently, with an absolute error, the merit function is: e2 ¼
nbval X
ðqi Rðhsi ; hvi ; Dusvi ; ki ; pÞÞ2
i¼1
if pj zpjmin and pj Vpjmax bja½1; m e2 ¼
nX bval
ðqi Rðhsi ; hvi ; Dusvi ; ki ; pðpj ¼pjmax Þ ÞÞ2 þ pj pjmax
i¼1
if pj zpjmax e2 ¼
nX bval
ðqi Rðhsi ; hvi ; Dusvi ; ki ; pðpj ¼pjmin Þ ÞÞ2 þ pjmin pj
modify the soil reflectance model or the free parameters, with the objective to improve the robustness of the inversion method. Any test provides the RMSE, the correlation coefficient and the coefficients a and b of the linear regression, for each free parameter. It gives also the RMSE of reflectance values: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u nbval uX u ðqi Rðhsi ; hvi ; Dusvi ; ki ; pÞÞ2 u t i¼1 : RMSE ¼ nbval The analysis was conducted using stand characteristics of the Fontainebleau forest (Dufreˆne et al., 1997). Free parameters were tree ground cover st and leaf density uf. Soil reflectance was set to qs = 5% in the green, qs = 7% in the red and qs = 8% in the near infrared. The initial number of simplexes was fixed to five. The merit function is a relative error with a threshold value qmin = 10%. The lookup table was calculated with 10 discrete values per free parameter, with sta[0, 1] and ufa[0.2 –0.8]. One hundred random values of white noise were applied to the simulated reflectance values, for each couple (st, uf). We could retrieve easily st (Table 2), at least if the understory has a low chlorophyll component. The situation differs for the retrieved uf. It is accurate if st is large and very inaccurate if st is low, which is not very important because LAI is proportional to the product of ufst. Generally speaking, with the assumption that a 5% proportional noise is reasonable, LAI and st can be accurately retrieved, provided that scene simulations are realistic. Actually, the imprecision on sensor calibrations and atmospheric corrections can imply a noise larger than 5%, in the absence of ground reference targets.
i¼1
if pj Vpjmin 5.5. Sensitivity analysis It is difficult to conduct a systematic assessment of the robustness of the inversion method because of the large number of factors: type of landscape, scene simulation, number and value of spectral channels, number and value of view directions, illumination configurations and free parameters. However, it is interesting to test if the inversion of simulated reflectance values leads to the input parameters used in order to simulate these reflectance values, even if some noise is added on these reflectance values. We considered two types of noise: an additional white noise and a proportional white noise. For example, with the proportional white noise, the inversion is performed on RkV = Rk + DRk, where ADRkA < ERk and E is the random error level (e.g., from 0% to 10%). This type of test is useful to fix some points of the inversion method such as the optimum number of simplexes and the threshold value that stops the inversion procedure. It can be helpful also to
5.6. Example of application: forest LAI mapping with SPOT images The inversion method was applied to SPOT images of the Fontainebleau forest (France) acquired from 1989 to
Table 2 Sensitivity analysis of the inversion procedure Additional white noise 0 LAI [0 – 9.3]
RMSE 0.087 r2 0.998 a 1.000 b 0.007 uf RMSE 0.066 [0.2 – 0.8] r2 0.8761 a 1.006 b 0.008 st RMSE 0.016 [0 – 100%] r2 0.997 a 1.007 b 0.001
Proportional white noise
0.01
0.05
0.05
0.1
0.770 0.845 0.927 0.184 0.267 0.049 0.317 0.354 0.122 0.842 0.967 0.038
1.031 0.738 0.888 0.238 0.301 0.004 0.101 0.466 0.167 0.737 0.961 0.044
0.455 0.945 0.983 0.023 0.162 0.400 0.737 0.133 0.052 0.968 0.986 0.013
0.723 0.869 0.975 0.052 0.210 0.196 0.565 0.216 0.080 0.926 0.97 0.026
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2000. Only major results are shown here. This work was part of the EMAC project (Dufreˆne et al., 1997) the overall scientific objective of which was to map forest LAI for each summer of each year because LAI is maximal at that period and as such is useful for initializing forest functioning models. In the frame of the work presented here, the objective was to investigate the possibility to obtain accurate LAI maps with a look-up table that was not designed for the view-sun configurations of the available SPOT images. First, SPOT images were atmospherically corrected with an iterative approach (Zagolski & Gastellu-Etchegorry, 1995) that corrects for environmental effects. The look-up table was calculated with 4-tree scenes and ellipsoidal tree crowns (15 m height, 10.5 m diameter) for the following discrete parameters: – three spectral leaf albedos that were directly computed from SPOT images, – three sun zenith angles (20j, 40j and 60j), – three soil reflectance values (0%, 10% and 30%), – three leaf density values (0.2, 0.4 and 0.8 m2/m3), – four tree ground covers (0%, 45%, 75% and 100%). Tree ground cover and leaf density were the free parameters used to assess forest LAI. The latter one was compared with in situ LAI measurements (Dufreˆne et al., 1997) in 38 deciduous stands and 13 coniferous stands. Coniferous stands have a larger RMSE than deciduous stands (Table 3). Total RMSE ranges from 1.1 to 1.5, with LAI stand values up to 9. These RMSE values can be explained by the inaccuracy of several factors or procedures such as atmospheric corrections and leaf optical properties. However, it must be noted that they are similar to the RMSE between in situ LAI and the LAI derived from a relationship such as LAI = f (NDVI), provided the latter one is calibrated with in situ LAI measurements. Much larger RMSE were obtained with relationships LAI = f (NDVI) that were not calibrated or that were calibrated for another view-sun configuration (Fig. 17). Here, the advantage of the look-up table inversion method is not to provide more accurate LAI values but to give acceptable results without using in situ LAI measurements. However, other in situ measurements (tree dimensions, etc.) were used to feed DART with realistic simulation
Table 3 RMSE between in situ LAI and the LAI derived from the inversion of SPOT images acquired between 1994 and 2000 Date
RMSE (deciduous)
RMSE (conifers)
RMSE (total)
10/07/94 10/07/95 17/07/96 13/08/97 06/08/98 21/07/00
1.36 1.51 1.28 1.48 1.50 1.21
0.79 0.92 0.72 0.86 0.83 0.75
1.25 1.40 1.18 1.36 1.38 1.12
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Fig. 17. Empirical analytical relationships LAI = f (NDVI) derived from the July 1996 SPOT image, for the 38 deciduous stands. NDVI is computed with SPOT channels XS2 and XS3.
scenes, which is necessary to simulate accurate reflectance values. We also investigated the usefulness of a complex 3-D model for the retrieval of LAI. For that, the 1996 SPOT image was inverted with a turbid reflectance model: the SAIL model (Verhoef, 1984). We used a simplex inversion method (Este`ve, 1998). This led to larger RMSE than those obtained with a 3-D reflectance model: 1.65 instead of 1.18. These worse results are mainly explained by the fact that the SAIL model does not take into account the architecture of the tree canopy. This neglect of canopy architecture explains also that SAIL derived LAI is smaller than the LAI derived from a 3-D reflectance model. Indeed, the impact of vegetation on satellite measurements is optimized if vegetation is homogeneously distributed as it is assumed with a turbid reflectance model such as SAIL. These results confirm that turbid reflectance models are less well adapted than 3-D reflectance models for simulating accurate reflectance values of landscapes with trees. In short, the inversion of SPOT images with the look-up table inversion method showed that the interpolation used in the inversion method allows us to obtain accurate LAI maps with a look-up table that was not specifically designed for the view-sun directions of the SPOT images. Thus, as expected, the method that was developed is efficient for generalizing a look-up table for any view-sun direction.
6. Concluding remarks Inversion methods of physically based vegetation reflectance models are efficient tools for extracting desired vegetation variables using directional and spectral data acquired by satellite borne sensors. Physically based models are potentially more effective than empirical models. However, the complexity of these models explains that important efforts are made to improve the
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robustness and accuracy of inversion methods. The two major inversion methods, i.e., neural network methods and look-up table methods, are characterized by a major drawback: they are not inherently designed to be easily generalized to any view-sun configurations. It implies that neural networks and look-up tables must be designed for all view-sun configurations used to acquire satellite images. This is an unacceptable constraint for operational applications. For example, it would require huge computational memory. The difficulty to generalize a pre-computed reflectance data base or an analytical expression of scene reflectance to any view-sun configuration arises from the complexity to interpolate on view and illumination angles. Interpolation on reflectance values around the hotspot configuration clearly illustrates this difficulty. Here, we present a method that generalizes a pre-computed look-up table to any view-sun configuration. An original point of the inversion method is that the look-up table is made of pre-computed coefficients of any analytical reflectance models that simulates accurately canopy reflectance for any view configuration. Here, it is a simple analytical model with six coefficients ci that depend on DART input parameters (i.e., surface variables and illumination configuration). Interpolation on these coefficients provides new coefficient values for any parameter value. Use of these new coefficient values in the analytical reflectance model gives reflectance values for any parameter value, provided the latter one is bounded by parameters used to compute the look-up table. Consequently, this approach has the potential for generalizing the look-up table inversion method for any view-sun configuration. For example, coefficients ci associated with a given sun direction (XV) are interpolated from coefficients ci that correspond to the sun directions that bound direction (XV). The inversion approach can work with look-up tables that are derived from any reflectance model. Here, we used the 3-D reflectance model DART. It must be noted that in presence of landscapes the BRF of which has a marked azimuth asymmetry, the look-up table must be built-up with scenes that are more complex than the 4tree scene. In that case, we must use an analytical reflectance model that can represent this possible azimuth asymmetry. After a successful theoretical test of robustness, the possibility to generalize a look-up table to any view-sun direction was successfully tested with the inversion of a series of SPOT images for determining the LAI and tree cover of a temperate forest (Fontainebleau). This led to LAI maps that are as accurate (i.e., RMSE c 1.3) as LAI maps that are derived from classical relationships that are calibrated with LAI in situ measurements. The advantage of the developed inversion method is to avoid the calibration step (i.e., collection of in situ LAI measurements), although some in situ data (e.g., tree character-
istics) must be collected, but not necessarily at the time of the satellite pass. Presently, work is being conducted in order to provide a reliable assessment of the accuracy of the inversion method. Acknowledgements The authors are grateful to the reviewers for their constructive remarks. They are also grateful to Eric Dufrene (Laboratoire d’Ecologie Ve´ge´tale, France) and his team for providing in situ LAI measurements. References Abuelgasim, A. A., Gopal, S., & Stralher, A. H. (1998). Forward and inverse modeling of canopy directional reflectance using a neural network. International Journal of Remote Sensing, 19, 453 – 471. Bacour, C. (2001). Contribution a` la determination des parame`tres biophysiques des couverts ve´ge´taux par inversion de mode`les de reflectance: Analyses de sensibilite´ comparatives et configurations optimales, Univ. Paris 7, PhD Thesis, 191 pp. Brent, R. P. (1973). Algorithms for minimization without derivatives. Englewood Cliffs, NJ: Prentice-Hall. Brodlie, K. W. (1977). In D. A. H. Jacobs (Ed.), The state of the art in numerical analysis. London: Academic Press, Chapter III.1. Cybenko, G. (1989). Approximations by superpositions of a sigmoidal function. Mathematical Control, Signal and Systems, 2, 303 – 314. Demarez, V. (1997). Mode´lisation du transfert radiatif et te´le´de´tection hyperspectrale pour le suivi temporel de la teneur en chlorophylle d’une foreˆt tempe´re´e. The`se, Universite´ Paul Sabatier Toulouse, III, 142 pp. Dufreˆne, E., Ledantec, V., Demarez, V., Gastellu-Etchegorry, J.P., Marty, G., Mougin, E., Proisy, C., Lacaze, B., & Rambal, S. (1997). Remote Sensing of the Fontainebleau Forest during Emac-94: Objectives and data collection program. Paris: Publications ESA. Este`ve, P. (1998). Inversion du mode`le de transfert radiatif DART, Paul Sabatier University, PhD Thesis, 147 pp. Gascon, F. (2001). Mode´lisation physique d’images de te´le´de´tection optique, Paul Sabatier University, PhD Thesis, 165 pp. Gascon, F., Gastellu-Etchegorry, J. P., & Lefe`vre, M. J. (2001). Radiative transfer model for simulating high-resolution satellite images. IEEE, 39(9), 1922 – 1926. Gastellu-Etchegorry, J. P., Demarez, V., Pinel, V., & Zagolski, F. (1996). Modeling radiative transfer in heterogeneous 3-D vegetation canopies. Remote Sensing of Environment, 58, 131 – 156. Gastellu-Etchegorry, J. P., & Gascon, F. (2002). An imaging VNIR and TIR ‘‘surface-atmosphere’’ 3-D radiative transfer model. Remote Sensing of Environment (in preparation). Gastellu-Etchegorry, J. P., Guillevic, P., Zagolski, F., Demarez, V., Trichon, V., Deering, D., & Leroy, M. (1999). Modeling BRF and radiation regime of tropical and Boreal forests—Part I: BRF. Remote Sensing of Environment, 68, 281 – 316. Gobron, N., Pinty, B., & Verstraete, M. M. (1997). Theoretical limits to the estimation of the leaf area index on the basis of visible and near-infrared remote sensing data. IEEE Transactions on Geoscience and Remote Sensing, 35(6), 1438 – 1445. Goel, N. S. (1988). Models of vegetation canopy reflectance and their use in estimation of biophysical parameters from reflectance data. Remote Sensing Reviews, 4, 1 – 213. Goel, N. S. (1989). Inversion of canopy reflectance models for estimation of biophysical parameters from reflectance data. Theory and applications of Optical Remote Sensing (pp. 205 – 251). New York: Ghassem Asrar. Wiley-Interscience.
J.P. Gastellu-Etchegorry et al. / Remote Sensing of Environment 87 (2003) 55–71 Goel, N. S., & Strebel, D. E. (1983). Inversion of vegetation canopy reflectance models for estimating agronomic variables: I. Problem definition and initial results using the Suits model. Remote Sensing of Environment, 13, 487 – 507. Goel, N. S., & Thompson, R. (2000). A snapshot of canopy reflectance models and a universal model for radiation regime. Remote Sensing Reviews, 18, 197 – 225. Goel, N. S., & Thompson, R. L. (1984). Inversion of vegetation canopy reflectance models for estimating agronomic variables: IV. Total inversion of the SAIL model. Remote Sensing of Environment, 15, 237 – 253. Guillevic, P., & Gastellu-Etchegorry, J. P. (1999). Modeling BRF and radiation regime of tropical and Boreal forests—Part II: PAR. Remote Sensing of Environment, 68, 317 – 340. Hornik, K., Stinchcombe, M., & White, H. (1989). Multi-layer feed-forward networks are universal approximators. Neural Networks, 2, 359 – 366. Jacquemoud, S., Bacour, C., Poilve´, H., & Frangi, J. P. (2000). Comparison of four radiative transfer models to simulate plant canopies reflectance: Direct and inverse mode. Remote Sensing of Environment, 74, 471 – 481. Kimes, D., Gastellu-Etchegorry, J. P., & Este`ve, P. (2002). Recovery of forest canopy characteristics through inversion of a complex 3D model. Remote Sensing of Environment, 79, 320 – 328. Kimes, D. S., Knyazikhin, Y., Privette, J. L., Abuelgasim, A. A., & Gao, F. (2000). Inversion methods for physically-based models. Remote Sensing Reviews, 18, 381 – 439. Knyazikhin, Y., Martonchik, J. V., Myneni, R. B., Diner, D. J., & Running, S. W. (1998). Synergistic algorithm for estimating vegetation canopy leaf area index and fraction of absorbed photosynthetically active radiation from MODIS and MISR data. Journal of Geophysical Research, 103, 32239 – 32256. Myneni, R. B., & Ross, J. (Eds.) (1991). Photon – vegetation interactions. New York: Springer Verlag, 565 pp. Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308 – 313. Pinel, V., & Gastellu-Etchegorry, J. P. (1998). Sensitivity of texture of high
71
resolution images of forest to biophysical and acquisition parameters. Remote Sensing of Environment, 65, 61 – 85. Pinty, B., Gobron, N., Widlowski, J. L., Gertsl, S. A. W., Vertsraete, M. M., Antunes, M., Bacour, C., Gascon, F., Gastellu-Etchegorry, J. P., Goel, N., Jacquemoud, S., North, P., Qin, W., & Thompson, R. (2001). Radiation transfer model intercomparison (RAMI) exercise. Journal of Geophysical Research, 106(D11), 11937 – 11956. Pinty, B., Verstraete, M. M., & Dickinson, R. E. (1989). A physical model of the bidirectional reflectance of vegetation canopies: Part II. Inversion and validation. Journal of Geophysical Research, 95, 11767 – 11775. Press, W. H., Teukolsky, A. A., Vetterling, W. T., & Flannery, B. P. (1992). Numerical recipes in C—the art of scientific programming. New York: Cambridge Univ. Press. Privette, J. L., Emery, W. J., & Schimel, D. S. (1996). Inversion of a vegetation reflectance model with NOAA AVHRR data. Remote Sensing of Environment, 58, 187 – 200. Quin, W., & Liang, S. (2000). Plane-parallel canopy radiative transfer modeling: Recent advances and future directions. Remote Sensing Reviews, 18, 281 – 305. Roujean, J. L., Leroy, M., & Deschamps, P. Y. (1992). A bidirectional reflectance model of the earth’s surface for the correction of remote sensing data. Journal of Geophysical Research, 97, 20455 – 20468. Twomey, S. (1977). Introduction to the mathematics of inversion in remote sensing and indirect measurements. Amsterdam: Elsevier Sci. Publish., Reprinted by Dover Publish., 243 pp. Verhoef, W. (1984). Light scattering by leaf layer with application to canopy reflectance modeling: The SAIL model. Remote Sensing of Environment, 16, 125 – 141. Wanner, W., Li, X., & Strahler, A. H. (1995, October 20). On the derivation of kernels for kernel-driven models of bidirectional reflectance. Journal of Geophysical Research, 100(D10), 21077 – 21089. Zagolski, F., & Gastellu-Etchegorry, J. P. (1995). Atmospheric corrections of AVIRIS images with a procedure based on the inversion of the 5S model. International Journal of Remote Sensing, 16(16), 3115 – 3146.
