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## Inversion of leaf optical properties study

The OPs of leaves reflect their combination of biochemical, morphological and physiological properties and play an essential role in many ecological and earth system processes. There are three principal methods for obtaining OPs of leaves: instrumental measurements, simulations of leaf biochemical parameters, and canopy spectral inversion.

**Optical instrument measurement**

Remote sensing simulations or inversions of leaf OPs require spectral measurements to verify accuracy. There are currently three main methods for measuring the OPs of leaves [38]: (1) The instrument’s optical fibre is connected to a port on an external integrating sphere, such as the LI-1800-12 or ASD RTS-3ZC, with an internal halogen light source for measuring the diffuse reflectance or transmittance. (2) The instrument’s optical fibre is connected to unique accessories (plant probes and leaf clips) with an internal halogen light source. Using a double-sided rotating head containing a recessed background plate, reflectance (black panel, reflectance < 5%) and transmittance (Spectralon white panel, reflectance > 99% in the visible to the near-infrared band) measurements can be carried out simultaneously. This measurement method has the advantage of reducing external disturbances such as angle of incidence or unstable light sources. (3) The instrument’s fibre optic is connected to a gun (Pistol Grip), and the leaf is observed vertically, for example, in the laboratory with light as the light source or the field with the sun at any angle of incidence as the light source. Then the bi-directional reflectance factor is measured, assuming that sensors and sunlight are assumed to be mono-directional.

Calibrating these data can be a daunting task, and most of the time, the calibration is not performed correctly. Few studies compare the reflectance of the same leaf recorded using different systems or spectrometers, each with a unique spectral resolution and sampling intervals. Castro-PhD Thesis, Université de Toulouse Esau et al. [39] perform an instrument comparison and standardization of sampling procedures. Depending on the instrument, they observe slight to severe differences in shape and amplitude between spectra of the same leaf. For some applications, such differences may not be significant at a practical level, as the calculated spectral indices are similar. For other applications, such as the use of physical models, such differences may be critical. Therefore, inverting PROSPECT model parameters using data measured by ASD probes or leaf clamps may lead to biased estimates of leaf biochemical parameters.

**Simulation of leaf radiative transfer**

Variations of leaf reflectance and transmittance are modelled by analyzing the interaction processes of electromagnetic waves with the biochemical components within the leaf and the leaf structure. Light propagation in plant leaves is mainly governed by absorption and scattering interactions. In parallel with advances in experimental measurements of the OPs of leaves, deterministic methods using different representations of the interaction of light with plant leaves have been developed. These models differ depending on the underlying physics and the complexity of the leaf. The simplest models treat the leaf as a single scattering and absorbing layer, while in the most complex models, all cells’ shape, size, location, and biochemical content are described in detail. Regardless of the approach used, these models have improved understanding of the interaction of light with plant leaves. Baranoski and Rokne [40], Ustin et al. [41], and Jacquemoud et al. [42] have extensively reviewed computer-based leaf models that have improved the understanding of the interaction between light and plant leaves from the late 1960s to the present. These models are divided into different categories and arranged in increasing complexity [43].

**Flat model**

The flat plate model, first proposed by Allen et al. [44] in 1969, treats a dense leaf as a homogeneous flat plate with a rough surface. Light incident on such a leaf surface is reflected and transmitted several times. It is partially reflected and partially transmitted at the first interface, and the transmitted part is then reflected back and forth between the two interfaces. The total reflectance of the plate can be obtained by summing the amplitudes of successive reflections and refractions. However, this model does not apply to non-compact leaves. After Allen et al. [45], Breece and Holmes [46] extended the flat plate model to non-compact leaves by introducing the generalized flat plate model, subdividing the leaf into N homogeneous compact plates separated by N-1 air spaces. In the 20th century, N values were extended from integers to the real number domain. The PROSPECT model, now widely used in the remote sensing community, was developed from the flat model [47]. It is one of the first radiative transfer codes to accurately model the hemispheric reflection and transmission of various plant leaves and conditions (monocotyledons, dicotyledons of healthy or senescent leaves) across the solar spectrum from 400 nm to 2500 nm. The input parameters for the early PROSPECT models are: structural parameter N, green pigment and water content to simulate fresh leaves. If dry leaf spectra need to be simulated, protein and cellulose + lignin content must be added. However, the plate model cannot be applied to needle leaves because they cannot be treated as discrete parallel plates.

**Compact spherical particle model**

designed a degree-of-freedom model precisely to calculate the OPs of dried and fresh slash pine (Pinus elliottii) needles to model the OPs of needles. The model treats needle cell structures as spherical cells with diameters and air gaps according to the laws of geometrical optics and then combines leaf thickness, absorption coefficients of water, chlorophyll, cellulose, lignin and nitrogenous compounds in the leaf to correct for the OPs of the needles.

**N flux model**

The N-flux model is derived from the Kubelka-Munk (KM) theory, which considers leaves plates filled with absorption and scattering coefficients [50]. In order to eliminate edge effects, lateral extension and boundary reflections below the plate are assumed to be absent, and the OPs are expressed as a function of the absorption coefficient, the scattering coefficient and the thickness of the leaf. The absorption coefficient and scattering coefficient are expressed in later versions as a function of the leaf thickness. Radiative transfer within the leaf is modelled using paired fluxes, such as two- and four-flux models. N-flux model has the advantage that the pigment content can be inverted without damage. The complete leaf biochemistry is described by Conel et al. [51], who used a two-flux model to study the effects of water, protein, cellulose, lignin and starch on mid-infrared radiation in leaves. However, they did not validate their model. Finally, note that the parameter N describing the internal structure of the leaf in the generalized flat plate model plays a role similar to that of the scattering coefficient in the KM model. One drawback of this approach is that it cannot consider the multiple scattering of leaf flesh objects of a size comparable to the wavelength of the incident radiation (cells, organelles, bubbles and others) and cannot characterize them.

**Radiative transfer equation**

The primary electromagnetic theory can describe the propagation of light through plant leaves. The leaf can be considered a random medium with a spatially varying dielectric constant, and the electric field variation can be described by Maxwell’s theory. Thus, the problem is simplified as the one-dimensional energy flow through the medium. In contrast to canopy radiative transfer, only a few models directly use the radiative transfer equations at the leaf scale. The mathematical complexity of the algorithm and the lack of information on the leaf’s internal structure and biochemical distribution lead to substantial simplifications, making this approach less efficient than more powerful formulations [52].

**Stochastic models**

Tucker and Garratt [53] present a primitive stochastic model in which the radiative transfer is modelled by a Markov chain, which is a stochastic process. They divide the leaf into two different tissues (palisade parenchyma and spongy mesophyll) and define states of photons (incident solar radiation, specularly reflected radiation, diffuse reflected radiation, diffuse transmitted radiation, absorbed radiation and scattered radiation, in each tissue). The random photon states can only take on discrete values, such as « absorbed in the palisade parenchyma » or « scattered in the spongy mesophyll ». The initial vector sets the incident radiation, and the OPs of the leaf is obtained when iterative state shifts are made until they smooth out.

#### Light tracing model

The advantage of ray-tracing models is that they can describe the complexity of the internal structure of a leaf (e.g., individual cells and their unique arrangement within the tissue). By defining the OPs of the leaf material (cell walls, cytoplasm, pigments, stomata, etc.), the propagation of individual photons incident on the leaf surface can be simulated using the laws of reflection, refraction and absorption. Once a sufficient number of rays have been simulated, a statistically valid estimate of the radiative transfer in the leaf can be deduced. The technique has been applied to many variants. The earliest studies were carried out at the cellular level. Senn [54], Haberlandt [55] and more recently, Gabrys-Mizera [56] and Bone et al. [57] construct geometric models of the passage of light through cross-sections of plant cells of different shapes, particularly epidermal cells, whose shape may influence the path of the incident light. Allen et al. [58] treat leaves as cell walls and cell gaps and model the spectrum of maple leaves using ray tracing. However, this method requires a significant computational load and has been commonly applied to validate simple models and understand the light transmission processes inside the leaf.

**Table of contents :**

**Chapter 1 Introduction **

1.1 Motivation

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

1.2.4 Problems

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

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

7.2 Innovations

7.3 Shortcomings and perspectives