Inversion of canopy radiative transfer
Taking the inversion of the OPs of leaves as an example, as the leaf and canopy radiative transfer models continue to be studied in-depth, researchers have found that the radiation signals observed by sensors can be used to interpret the biophysical state of the vegetation. The canopy radiation signal captured by the sensor can be simulated by a priori knowledge of the vegetation (canopy structure, leaf OPs, etc.) using a suitable canopy radiative transfer model, a process known as forwarding simulation. In contrast, the bio-optics of the leaf or the canopy’s geometry can be obtained by using a suitable canopy radiative transfer model based on the a priori knowledge of the radiation signal captured by the sensor. For example, Otterman  assumed a canopy structure and inverted a protrusion model to determine the reflectance of the leaf in the nadir observation direction and several bands. Otterman  later attempted to invert the leaf orientation but found it difficult to isolate the leaf reflectance from the leaf orientation or LAI. Kuusk  and Privette et al.  invert the OPs of leaves and the spatial distribution of scatters in the canopy by inversion of a bi-directional reflectance analysis model.
A major problem limiting the accuracy of the inversion is multi-solvability. An accurate canopy radiative transfer model is difficult to invert the input parameters from the output parameters, and it is difficult or even impossible to find the inversion function corresponding to the radiative transfer equation. Inverting the model is to find the parameter that minimizes the difference between the measured and simulated data. The critical factor for the success of leaf OPs inversion is choosing a suitable canopy radiative transfer model and inversion method. Commonly used canopy radiative transfer models and their applicability are listed in section 1.2.1, and standard inversion methods are described in detail below, including iterative optimization algorithms [47, 63], Look-Up Tables algorithms (LUTs) [64, 65], and machine learning algorithms such as neural networks tool (NNT) and support vector machines [66-68].
Iterative optimization algorithm
An iterative optimization algorithm is a process that continually recurses old values of a target parameter (e.g., the OPs of a leaf) to new values, provided that the initial values of the target parameter are known. The iteration can be done by setting different iteration steps to run the radiative transfer model repeatedly to find the most suitable value of the OPs of the leaf for the simulated and measured spectral values of the canopy. Standard methods include genetic algorithms, quasi-Newtonian methods and least-cost functions. Iterative optimization algorithms take full advantage of the computer, such as high computing speed and repetitive operations. However, iterative optimization algorithms also have limitations: (1) the initial value strongly influences the final inversion result. In addition, if the model is not monotonic, the inversion result may fall into a local optimum rather than a global optimum. (2) The error function significantly impacts the final inversion results. In addition, different error functions may require different a priori knowledge (e.g., spectral parameters, structural parameters). (3) It is computationally intensive and unsuitable to equally computationally intensive canopy radiative transfer models such as ray tracing. Despite these limitations, iterative optimization algorithms are still the most classical inversion methods. Many authors used iterative optimization to obtain biochemical parameters for various vegetation types [47, 63, 69].
Look-Up Tables method
The use of iterative optimization algorithms for bio-optical property inversion with remote sensing from large vegetation areas requires many cost functions and a high operational load, which is a problem that can be well avoided with LUTs. LUTs are an efficient and straightforward inversion method. LUTs can improve the problem of machine learning algorithms falling into local optima when the training sample is small. In addition, LUTs can reduce the outliers between the simulated spectra and the observed spectra of the sensors. Moreover, LUTs do not rely on ground sampling sites for modelling, so they can invert vegetation parameters based on radiative transfer models when the number of ground sampling sites is insufficient, effectively eliminating the instability of statistical methods. However, the LUT size can be proliferated if the step size of the input parameters is too small, leading to non-representative results. Therefore, a parameter sensitivity analysis should first be conducted to set reasonable parameters’ range and step size before building a LUT to improve the efficiency and accuracy of the inversion. Darvishzadeh and Matkan  obtain canopy chlorophyll content using a LUT approach by inverting the PROSPECT + SAIL (PROSAIL) vegetation radiative transfer model. Dorigo  uses Compact High Resolution Imaging Spectrometer (CHRIS) / Project For On-Board Autonomy (PROBA) multi-angle remote sensing images to obtain the chlorophyll content of the canopy.
Machine learning methods
The goal of machine learning is to draw lessons from training samples. The main machine learning methods for the inversion of vegetation bio-OPs are currently artificial neural networks and support vector machines. The inversion based on artificial neural networks consists of two main steps: (1) training the neural network using many training samples; and (2) using the training network to invert the canopy bio-optical property. Neural networks have a solid non-linear mapping capability and are suitable for complex internal mechanisms problems. However, neural networks require a large number of training samples in the training process and are prone to overfitting. The support vector machine approach is mainly applied in remote sensing for classification and less frequently in vegetation bio-optical parameter inversion. Durbha  uses support vector machines and the PROSAIL radiative transfer model to invert the leaf area index from Multi-angle Imaging SpectroRadiometer (MISR) images with reasonable accuracy. Therefore, the applications of machine learning methods to invert vegetation biochemical parameters needs further research. A short comparison between the look-up tables method and machine learning methods is that machine learning methods do not need to know the underlying data distribution explicitly. Hence, machine learning methods are developed without assuming a particular probability density distribution, which is why they work well with all kinds of data types . However, for the lookup table methods, the data distribution must be considered when creating the lookup table. Machine learning methods also offer the possibility of incorporating prior knowledge and the flexibility to include different data types in the analysis. The machine learning methods use all available training data. As a result, all data take effect. However, for the LUT method, not all data take effect. The main problem for machine learning is the collinearity complicating regression. However, noise removal is still a standard and much-needed pre-processing step for the lookup table method. Although various inversion methods are used, at present, the research on the inversion of leaf spectral characteristics is still based on the assumption that the scene is a homogeneous canopy. This assumption may be reasonable for dense vegetation cover areas such as forests. However, this assumption may not be feasible for urban plants, especially in the current commonly used satellite images (i.e., 10 m for Sentinel-2). Therefore, one objective of our study is to retrieve leaf optical property (OP) in the urban areas based on DART radiative transfer model and iterative optimization algorithm.
Scattering by Arbitrary Inclined Leaves (SAIL) model
Canopy radiation characteristics depend primarily on canopy structure. Different modelling techniques reflect the structure of the canopy at different scales. In the radiative transfer model, the vegetation canopy can be assumed to be a one-dimensional mixed medium (e.g. atmosphere and water), and the canopy units are randomly distributed, as shown in Figure 2.1. The canopy is infinitely uniform horizontally for the one-dimensional case with a limited vertical variation. The substrate of the canopy may be soil or another canopy (e.g. grass), and for simplicity, the canopy may be considered foliage only because the main photosynthetic tissue in the vegetation is leaves. When radiation is transmitted in the vegetation, it is more likely to interact with the leaves to change the radiation characteristics. The average effect of proportional distribution can summarise the true canopy consisting of leaves and branches. The physical properties of the leaf include leaf size, leaf orientation, leaf surface roughness, and leaf optical properties such as reflectivity, transmittance, and absorptivity. Since we pay more attention to the overall nature of the leaves, it is necessary to define some vegetation group characteristic parameters. They are a refined description of the vegetation canopy structure and optical properties and are the results of the statistical average of the distribution of all leaves. The main structural parameters of the canopy include the leaf area index (LAI) and the leaf inclination angle distribution function (LAD).
SAIL model radiative transfer simulation
SAIL (light scattering by arbitrarily inclined leaves) model is extended from Suits model, which assumes that the canopy consists only of vertical and horizontal leaves and parameterizes canopy structure, sun, and observation geometry. The main difference between the SAIL model and the Suit model is that leaves replace the horizontal and vertical projections of the Suit model at arbitrary angles close to reality. Results show that the odd inflexion point disappears compared with the Suits model.
The input parameters of the SAIL model include optical properties of leaves (i.e., reflectance and transmittance) and soil (i.e., reflectance), view geometry (sun zenith angle, view zenith angle, and relative azimuth angle between sun azimuth angle and view azimuth angle), Leaf area index (LAI), the average leaf angle (ALA). The output parameter is the canopy reflectance. The computer simulation code is open-source and provided by several language versions on the website (http://teledetection.ipgp.jussieu.fr/prosail/).
The continuous vegetation radiative transfer equation is a physical condition describing the value of the radiation radiance in any direction at any point within the vegetation layer that should be satisfied, together with the boundary conditions. This differential-integral equation is in principle solvable, requiring that the solution of such an equation must satisfy: (i) the determination of the phase function; and (ii) the determination of the boundary conditions. So far, it has not been possible to find a strictly analytical solution of the RT equation, only a variety of approximate solutions, each of which has its own adaptation conditions.
Discrete Anisotropic Radiative Transfer Model
The DART Model (https://dart.omp.eu/#/) is a physically-based 3D radiative transfer model that simulates the radiative transfer of electromagnetic radiation from visible to thermal infrared wavelengths in a realistic 3D structural scene. It has been developed at the CESBIO in Toulouse, France, since 1992 and was patented in 2003. It provides radiation balance and measurements from satellite, airborne and ground-based passive (imaging spectrometer) to remote sensing active light detection and ranging (LiDAR). The DART [28, 88, 103-105, 129-133] modelled the surface explicitly, so the spatial extent and accuracy of the simulation are limited only by the input data and available computational power.
DART scene creation
DART scenes can be built with both voxels or facets, and the voxels representing vegetation in the DART model can be filled in two ways to represent the scattering properties, including turbid media or discrete triangular facets. A turbid medium is the light scattering of significant intensity due to irregular (randomly distributed) optical inhomogeneities in composition or generation. Scattering in a turbid medium causes a change in the initial direction of the light striking the medium. Whether using the turbid medium or triangular facets to fill the canopy depends primarily on the size of the trees relative to the resolution of the simulated scene (i.e., the voxels size that make up the scene). The triangular facets can be independent of the unit voxels that make up the scene, whereas the turbid medium depends on the unit voxels. For example, if the size of the simulated vegetation is much smaller than the resolution of the simulated image (i.e., the size of the constituent scene voxels), then it is preferable to use triangulated surfaces for filling. If the size of the simulated vegetation is much larger than the resolution of the simulated image, then both are possible. When the scene resolution is too large, and the canopy size is too small, care needs to be taken when filling with turbid media because no corresponding vegetation voxel can be created when the foliage makes up less than 50% of the unit voxel.
DART supports three approaches to modelling the vegetation in a scene. The first is an entirely homogeneous ideal voxel consisting of a turbid medium or triangular surface, referred to as a homogeneous assumption canopy. The second is a tree with a simple geometric shape (which includes a canopy modelled by a voxel filled with a turbid medium or a discrete triangular surface), and the vegetation scene constructed in this way is said to be a simplified heterogeneous canopy. The third is a three-dimensional real structural canopy composed of facets, and the vegetation scenes constructed in this way are called real structural canopies. All three vegetation modelling approaches are addressed in this study.
The simplified heterogeneous canopy scene creates a tree consisting of two parts: the trunk and the crown. The concept of « species » is used to manage the different tree types. Trees of the same species share some of the same characteristics, such as the way the canopy is modelled, the OP of the tree, the vertical hierarchical stratification of the defined canopy, the density of the leaf area volume, the size and shape of the canopy. The trunks are vertically stacked in an octagonal shape, both within or below the canopy. The crown can be defined in different shapes: ellipsoid, compound ellipsoid, truncated cone, trapezoid, and compound cone. For canopies with hierarchical layers in the vertical direction, the relative height of the layer, the relative leaf area voxel density of the layer, the relative trunk diameter, the OP of the leaves and trunk, the proportion of clumping in the horizontal direction, and so on can be defined on each hierarchical layer. There are two types of infill methods within the canopy. The first method is the turbid medium canopy layer, which is composed of a turbid medium and void voxel. In the DART model, a voxel is created as a turbid medium voxel only when the percentage of leaves within the voxel reaches 50% or more of the total voxel. The user specifies the parameters that make up the turbid medium voxels, such as the leaf area index or leaf area voxel density, the branch area index, the leaf inclination index, etc. In addition, the proportion of the whole leaf voxels occupied by the leaf cannot exceed 100%, and any portion exceeding 100% will be preserved in the surrounding neighbourhood of other voxels. The second infill method is a discrete isosceles triangular facet, where the triangle’s height is twice the height of the bottom edge, and the user can specify the total area or the total number of triangles within the canopy. It has been shown that whether or not short branches are simulated within the canopy has little effect on the total BRF . Similar filling rules can also be applied to the creation of vegetation plots.
Table of contents :
Chapter 1 Introduction
1.2 Research background
1.2.1 Canopy reflectance simulation study
1.2.2 Inversion of leaf optical properties study
1.2.3 Inversion of leaf area index study
1.3 Outline of the thesis
1.3.1 Study content
1.3.2 Thesis structure
Chapter 2 Basics of vegetation canopy radiative transfer models
2.1 Radiative transfer models
2.1.1 Scattering by Arbitrary Inclined Leaves (SAIL) model
2.1.2 Discrete Anisotropic Radiative Transfer Model
2.1.3 Radiosity-graphics based model
2.2 Simulating Canopy BRF with Radiosity-Graphics based Model (RGM) at Pixel Scale based PhD Thesis, Université de Toulouse on OpenACC
2.2.1 Reflectance simulation of realistic structural single tree canopy
2.2.2 Reflectance simulation of simplified heterogeneous canopy scene
Chapter 3 Continuous-time phase reflectance simulation of the realistic structural maize scene
3.1 3D maize scene modelling
3.2 Multi-temporal simulation analysis of maize canopy reflectance
Chapter 4 Inversion of leaf optical properties in urban areas at the sub-pixel scale
4.1 Inversion of leaf optical properties using simulated images
4.1.1 Noiseless ideal case for leaf optical properties inversion
4.1.2 Artificially noise case for leaf optical properties inversion
4.2 Inversion of leaf optical properties using satellite image
4.2.1 Study area overview and PlanetScope data pre-processing
4.2.2 Inversion of optical properties of leaves from multispectral images
Chapter 5 Using vegetation indices to estimate leaf area index
5.1 Using negative soil adjustment factor of SAVI to mitigate vegetation index saturation effect
5.1.1 Study of index isoline and vegetation isoline
5.1.2 Relationship between vegetation indices and LAI
5.2 Using hotspot signature vegetation indices to estimate LAI
5.2.1 Evaluation of inversion of leaf area index without noise interference
5.2.2 Evaluation of inversion of leaf area index with random noise interference
Chapter 6 Analysis of the influence of environmental noise on vegetation indices
6.1 Comparison of vegetation indices for soil noise resistance
6.2 Effects of atmosphere on vegetation indices
6.3 Effect of spectral response functions on vegetation indices
6.3.1 Pre-calibration error assessment
6.3.2 Band correlation coefficient calibration
6.3.3 Post-calibration error assessment
Chapter 7 Conclusion and perspectives
7.1 Major conclusions
7.3 Shortcomings and perspectives