radiative-transfer-models.qmd

# Radiative transfer models {#sec-RTM}

Radiative transfer describes the energy exchange through electromagnetic radiation, with varying influences from different chemical compounds [@verrelst2017global].
Radiative transfer models leverage this physical knowledge by simulating energy transfer from plant leaves or vegetation canopies, enabling the retrieval of vegetation properties from remotely sensed data [@yang2020spart].
Compared to machine learning approaches, radiative transfer models provide enhanced generalizability and robustness, as they embody a physical process rather than merely extracting a signal from data [@berger2018evaluation].

![This image depicts an anonymous artist's conception of cross section of a maple leaf on the left, and a graphical illustration of a simple radiative transfer model on the right. The model consists of 100 circular arcs simulating the scattering of light through the leaf. The illustration was originally published by @allen1973willstatter.](resources/PlateModel.png){#fig-PlateModel}

## Leaf-level radiative transfer models

One of the most prevalent radiative transfer models is [Prospect]{.smallcaps} (leaf optical [pro]{.underline}perties [spect]{.underline}ra), initially introduced by @jacquemoud1990prospect.
Drawing upon the plate model by @allen1969interaction, it simulates both reflectance and transmittance of a plant leaf across wavelengths ranging from 400 to 2500 nm.
The first version of the [Prospect]{.smallcaps} model featured only three input variables: leaf structure ($N$), leaf chlorophyll content ($c_{a+b}$), and leaf water content ($c_w$).
Based on that, the model returns the expected reflectance ($\boldsymbol \rho$) and transmittance ($\boldsymbol \tau$) for each simulated wavelength ($\lambda$).

$$f: [N, c_{a+b}, c_w] \in \mathbb R^{3} \rightarrow [\boldsymbol \rho, \boldsymbol \tau] \in \mathbb R^{2\times 2101}$${#eq-prospect}

Over the past four decades, the initial [Prospect]{.smallcaps} model has undergone several updates to encompass leaf mass, tannins, anthocyanins, and carotenoids [@fourty1996leaf; @jacquemoud1996estimating; @baret1997estimation; @fourty1998spectral].
For a long period, protein and other non-protein carbon-based compounds were consolidated into a single variable due to the challenge of disentangling them [@jacquemoud2000comparison].

In a notable development, the newest iteration of the [Prospect]{.smallcaps} model, [Prospect-Pro]{.smallcaps}, @feret2021prospect successfully disentangled these components by splitting the earlier used variable of leaf mass into protein content and other carbon-based content, which include cellulose, lignin, hemicellulose, starch, and sugars.
[Prospect-Pro]{.smallcaps} was calibrated and validated using the leaf optical properties experiment ([Lopex]{.smallcaps}) dataset, which comprises measurements from over fifty plant species collected in northern Italy [@hosgood1995leaf].
This model incorporates seven distinct classes of absorbing compounds, collectively referred to as vector $\mathbf c$.

$$f: [N, \mathbf c, \alpha] \in \mathbb R^{9} \rightarrow [\boldsymbol \rho, \boldsymbol \tau] \in \mathbb R^{2\times 2101}$${#eq-prospectpro}

[Prospect-Pro]{.smallcaps} also enables the adjustment of the solid angle $\alpha$ for incident light on the leaf surface, initially determined to be $\alpha = 60^\circ$ by @jacquemoud1990prospect using a limited dataset.
Subsequently, @feret2008prospect discovered that $\alpha = 40^\circ$ was a more accurate value.
An overview of the parameters used in [Prospect-Pro]{.smallcaps} can be found in [table @tbl-prospect].

+----------------+------------------+------------------------------------------------------+------------------------+
|**Abbreviation**|                  | **Parameter**                                        | **Unit**               |
+================+==================+======================================================+========================+
| $N$            | `N`              | Leaf mesophyll structure parameter                   |                        |
+----------------+------------------+------------------------------------------------------+------------------------+
| \              | `CHL`\           | Chlorophyll content\                                 |$\mu \text{g cm}^{-2}$\ |
| \              | `CAR`\           | Carotenoid content\                                  |$\mu \text{g cm}^{-2}$\ |
| \              | `ANT`\           | Anthocyanin content\                                 |$\mu \text{g cm}^{-2}$\ |
| $\mathbf c$    | `BROWN`\         | Tannin content\                                      |$-$\                    |
|                | `EWT`\           | Equivalent water thickness\                          |$\text{g cm}^{-2}$\     |
|                | `PROT`\          | Protein content\                                     |$\text{g cm}^{-2}$\     |
|                | `CBC`            | (Non-protein) carbon-based constituent content       |$\text{g cm}^{-2}$      |
+----------------+------------------+------------------------------------------------------+------------------------+
| $\alpha$       | `alpha`          | Solid angle for incident light at surface of leaf    |$^{\circ}$              |
+----------------+------------------+------------------------------------------------------+------------------------+

: Input variables of the [Prospect-Pro]{.smallcaps} model as documented by @feret2022package. The rows 2 - 7 show the elements of the constituents vector $\mathbf c$. {#tbl-prospect tbl-colwidths="[5,20,90,20]"}

The [Prospect]{.smallcaps} calculations rely on well-established laws of physics, with the Fresnel equations serving as the foundation.
These equations describe the reflection and transmission of electromagnetic radiation between distinct optical media [@fresnel1821].
Both $s$ and $p$ polarizations are included in the equations, which necessitate a refractive index $n$ --- assumed to be $n > 1$ for the leaf and $n = 1$ for air --- and an angle of incidence $\theta$.

$$t_s = \frac{4\sqrt{1-\sin^2 \theta} \sqrt{n^2-\sin^2 \theta}}{\left(\sqrt{1-\sin^2 \theta} + \sqrt{n^2-\sin^2 \theta}\right)^2}
\quad
t_p = \frac{4n^2\sqrt{1-\sin^2 \theta} \sqrt{n^2-\sin^2 \theta}}{\left(n^2\sqrt{1-\sin^2 \theta} + \sqrt{n^2-\sin^2 \theta}\right)^2}
$${#eq-fresnel}

As the focus lies on the transmissivity for diffuse light originating from all angles $\theta$ between 0 and $\frac{\pi}{2}$, it is necessary to integrate across all possible angles to obtain the average transmissivity $\overline{T}$.

$$\overline{T} = \frac{\int_0^{2\pi} d\varphi \int_0^{\alpha} t(\theta,n) \cos \theta \sin\theta d \theta}{\int_0^{2\pi} d\varphi \int_0^{\alpha}  \cos \theta \sin\theta d \theta}$${#eq-TAVunintegrated}

The integration for the special case where the maximum incidence angle $\alpha$ equals $\frac{\pi}{2}$ was performed by @stern1964transmission.
Subsequently, @allen1973transmission conducted the integration of the average transmissivity $\overline{T}$ for a generalized angle of incidence $\alpha$.
This approach is currently employed in the [Prospect-Pro]{.smallcaps} computation [@feret2022package].
The expression for average transmissivity is rather complex, as to be seen in [equation @eq-TavTsTp].
A visual representation of how the average transmissivity varies as a function of the refractive index $n$ and the angular range $\alpha$ can be found in [figure @fig-SAC], displayed at the bottom.

![The [Prospect]{.smallcaps} model simplifies a leaf's structure by representing it as a flat plane, rather than accounting for its complex three-dimensional form, which consists of various components and a textured surface (**A**, created using content by @wheeler2011). In the sketch inspired by @jacquemoud1992utilisation (**B**), the different angles incorporated in [equation @eq-TAVunintegrated] are illustrated: the general angles for azimuth ($\varphi$) and zenith ($\theta$) as well as the angle of maximum incident ($\alpha$). In addition to that, the arrows indicate the reflectance ($\rho$) and transmittance ($\tau$).](resources/LeafStructure.png){#fig-LeafStructure}

$$
\begin{aligned}
T_s(\alpha, n) =
\frac{1}{96}\frac{(n^2-1)^4}{\left[\sqrt{\left(\sin^2 \alpha -\frac{1}{2} (n^2 + 1)\right)^2 -\frac{1}{4} (n^2 - 1)^2} - \sin^2 \alpha +\frac{1}{2}  (n^2 + 1)\right]^3} -
\\\
\frac{1}{4}\frac{n^4 - 2n^2 +1}{\sqrt{\left(\sin^2 \alpha -\frac{1}{2} (n^2 + 1)\right)^2 -\frac{1}{4} (n^2 - 1)^2} - \sin^2 \alpha + \frac{1}{2} (n^2 + 1)} -
\\\
\frac{1}{2} \left( \sqrt{\left(\sin^2 \alpha -\frac{1}{2} (n^2 + 1)\right)^2 -\frac{1}{4} (n^2 - 1)^2} - \sin^2 \alpha + \frac{1}{2} (n^2 + 1) \right) +
\\\
\frac{1}{12} \frac{(n^2 - 1)^4}{(n + 1)^6} + \frac{1}{2} \frac{(n^2-1)^2}{(n+1)^2} + \frac{1}{4} (n^2+2n+1)
\end{aligned}
$$

$$
\begin{aligned}
T_p(\alpha, n) =
\frac{-2 n^2}{(n^2 + 1)^2} \left[ \sqrt{\left(\sin^2 \alpha - \frac{1}{2} (n^2 + 1)\right)^2 - \frac{1}{4} (n^2 - 1)^2} - \sin^2 \alpha + \frac{1}{2} (n^2 + 1) - \frac{1}{2} (n + 1)^2 \right] -
\\\
\frac{2  n^2  (n^2 + 1)}{(n^2-1)^2}  \cdot \ln \left[ \sqrt{\left(\sin^2 \alpha - \frac{1}{2} (n^2 + 1)\right)^2 - \frac{1}{4} (n^2 - 1)^2} -
\frac{\sin^2 \alpha - \frac{1}{2}(n^2 + 1)}{(n+1)^2}
\right] +
\\\
\frac{1}{2} n^2 \left\{ \left[ \sqrt{\left(\sin^2 \alpha - \frac{1}{2} (n^2 + 1)\right)^2 - \frac{1}{4} (n^2 - 1)^2} - \sin^2\alpha - \frac{1}{2} (n^2 + 1)\right]^{-1} - \frac{2}{(n + 1)^2} \right\} +
\\
\frac{16n^4 (n^4 + 1)}{(n^2+1)^3(n^2-1)^2} \cdot \ln \left[\frac{
2 (n^2 + 1)  \left(\sqrt{\left(\sin^2 \alpha - \frac{1}{2} (n^2 + 1)\right)^2} - \sin^2 \alpha + \frac{1}{2}  (n^2 + 1)\right)
- (n^2 - 1)^2}{(n^2 + 1)(n + 1)^2  - (n^2-1)^2} \right] +
\\\
\frac{16 n^6}{(n^2 + 1)^3} \left[
{2 (n^2 + 1)  \left(\sqrt{\left(\sin^2 \alpha - \frac{1}{2} (n^2 + 1)\right)^2} - \sin^2 \alpha + \frac{1}{2}  (n^2 + 1)\right)
- (n^2 - 1)^2} \right]^{-1} -
\\\
\frac{16n^6}{(n^2 + 1)^3\left( (n^2 + 1)   (n + 1)^2 - (n^2 - 1)^2 \right)}
\end{aligned}
$$

$$
\overline{T}(\alpha, n) = \frac{T_s(\alpha, n) + T_p(\alpha, n)}{2 \cdot \sin^2 \alpha}
$${#eq-TavTsTp}

In the [Prospect-Pro]{.smallcaps} model, reflectance and transmittance depend on multiple variables, as well as numerous fixed, empirically estimated parameters.
These parameters encompass the refractive index ($n$) and specific absorption coefficients ($s$), which all are individually fitted for each absorbing compound in $\mathbf c$ across a diverse range of plant species.
Additionally, these parameters are estimated for every wavelength $\lambda$ between 400 and 2500 nm.
As a result, [Prospect-Pro]{.smallcaps} relies on a total of 8 × 2101 = 16'808 empirically estimated internal parameters.
With these parameters in hand, the overall absorption coefficient can be computed as follows.

$$k_\lambda = \frac{1}{N} \sum_i c_i s_{i,\lambda}$${#eq-k}

Here, $c_i$ represents the content of the $i$^th^ constituent in the vector $\mathbf c$ (as detailed in [table @tbl-prospect]), and $s_i$ denotes the specific absorption coefficient of the $i$^th^ constituent and the $\lambda$^th^ wavelength.
Although the computation of reflectance and transmittance is not linear with respect to $k$, it is still possible to discern the bandwidths in which a particular compound's presence would have an impact based on the specific absorption coefficients $s$ displayed in [figure @fig-SAC].

![*Top:* This figure shows the refractive index (**n**) and the eight specific absorption coefficients ($s$) for different constituents of the [Prospect-Pro]{.smallcaps} model as functions of the wavelength $\lambda$. This encompasses the parameters chlorophyll content (**CHL**), carotenoid content (**CAR**), anthocyanin content (**ANT**), brown pigment content (**BROWN**), equivalent water thickness (**EWT**), leaf mass per area (**LMA**), protein content (**PROT**), and (non-protein) carbon-based constituents content (**CBC**). *Bottom:* The average transmissivity $\overline{T}$ of a plane dielectric surface for different values of the refractive index $n$ over the angular range $\alpha$. Adapted from @jacquemoud1990prospect.](resources/SpecPROSPECT.png){#fig-SAC}

Utilizing the overall absorption coefficient $k$, the transmission for isotropic light $\varphi$ can be calculated for each wavelength $\lambda$ with $k = k_\lambda$ using the upper incomplete gamma function in accordance with the Beer-Lambert law [@lambert1760photometria; @beer1852bestimmung].

$$\varphi = (1 - k) \cdot e^{-k} + k^2 \cdot \Gamma(k) = \\
(1 - k) \cdot e^{-k} + k^2 \cdot \int_k^\infty \frac{e^{-t}}{t} dt$${#eq-phi}

In the case of [Prospect-Pro]{.smallcaps}, the upper incomplete Gamma function $\Gamma(k)$ is simplified and computed according to @expint.
Using this information, the final equations for transmittance $\tau$ and reflectance $\rho$ can be expressed (@eq-rhoAndTau).
Due to space limitations, the substituting terms $A, B, C, D, E, F$, and $G$ are employed (@eq-substitutions).

$$\begin{aligned}
A = \frac{ \varphi \: \overline T(90, n)} { n^2\left[1 - \varphi^2 \: \left(1 - n^{-2} \: \overline T(90, n) \right)^2\right] }
\qquad
B = \overline T(90, n) \: A
\\[15pt]
C = 1 - \overline T(90, n) + \varphi \: B \: \left(1 - \frac{\overline T(90, n)}{n^2}\right)
\\[15pt]
D = \sqrt{(1 + C + B) \: (1 + C - B) \: (1 - C + B) \: (1 - C - B)}
\qquad
E = \frac{1 + C^2 - B^2 + D}{2\,C}
\\[15pt]
F = \frac{1 - C^2 + B^2 + D}{2\,B}
\qquad
G = F^{N-1}
\qquad
H = \frac{E \, (G^2 - 1)}{E^2 \, G^2 - 1}
\end{aligned}$${#eq-substitutions}

$$\rho = 1 - \overline T(\alpha, n) \left[ 1 + \varphi \, A \left(1 - \frac{\overline T(90, n)}{n^2}  \right) + \frac{A \, H \, B}{1 - H \, C}  \right]
\quad
\tau =  \frac{\overline T(\alpha, n) \, A \, G \, (E^2 - 1)}{(E^2 \, G^2 - 1) \, (1 - H \, C)}$${#eq-rhoAndTau}

The impact of distinct compounds on absorption is demonstrated in [figure @fig-ProspectModelOutput].
This figure vividly portrays the absorption of electromagnetic radiation by proteins in the short-wave-infrared spectrum, specifically within the range of 1500 to 2500 nm wavelength.
In contrast to carbon-based compounds, proteins exhibit no absorption in the near-infrared or visible light spectra.

![This figure illustrates the output of the [Prospect-Pro]{.smallcaps} model. Every plot shows an overlay of 20 model runs where only one variable is changing *ceteris paribus*. The three variables illustrated are chlorophyll content (**A**: `CHL` $\in [10,70]\; \mu\text{g} \: \text{cm}^{-2}$), (non-protein) carbon-based content (**B**: `CBC` $\in [0,0.02] \; \text{g} \: \text{cm}^{-2}$), and protein content (**C**: `PROT` $\in [0,0.02] \; \text{g} \: \text{cm}^{-2}$).](resources/ProspectModelOutput.png){#fig-ProspectModelOutput}

## Canopy-level radiative transfer models

The [Prospect]{.smallcaps} model is specifically designed to simulate the reflectance and transmittance of an individual leaf.
However, simulating the reflectance of an entire canopy proves to be far more intricate.
Canopy reflectance is affected by numerous factors such as leaf inclination, solar irradiance angle, viewing angle of the canopy, and the interference of soil reflectance as well as atmospheric effects.
To effectively account for these diverse variables, substantial extensions are necessary, which ultimately results in a significantly more complex simulation process.
Consequently, this section outlines several approaches for utilizing the [Prospect]{.smallcaps} model at the canopy level.

### One-dimensional canopy reflectance models

The radiative transfer model [Sail]{.smallcaps} ([s]{.underline}cattering by [a]{.underline}rbitrary [i]{.underline}nclined [l]{.underline}eaves) was one of the earliest models to simulate the bidirectional reflectance of a canopy.
[Sail]{.smallcaps} makes reflectance a function of both solar incident and viewing angles [@verhoef1984light; @verhoef1985earth].
As implied by its name, [Sail]{.smallcaps} represents the canopy through the simulation of numerous randomly inclined leaves situated above the soil.
These leaves are assumed to be uniformly distributed in space but inclined according to a specific leaf inclination angle distribution function.

The most recent iteration of the [Sail]{.smallcaps} model is known as [4Sail]{.smallcaps}.
While it simulates the same process as the original version, [4Sail]{.smallcaps} is more numerically robust and optimized for enhanced speed [@verhoef2007unified].
The parameters used in the [Sail]{.smallcaps} model can be found in [table @tbl-SAIL].

+------------------------+------------------------------------------------+-------------------------------+
|                        |  **Parameter**                                 | **Unit**                      |
+:=======================+:===============================================+:==============================+
| `lai`\                 | Leaf area index\                               |$\text{m}^2 \; \text{m}^{-2}$\ |
| `q` \                  | Hot spot parameter\                            |$\text{m} \; \text{m}^{-1}$\   |
|$\rho_{\text{soil}}$ \  | Soil reflectance\                              |$\%$\                          |
|$\alpha_{\text{soil}}$\ | Soil brightness factor\                        |\                              |
| `skyl`       \         | Fraction of diffuse illumination\              |\                              |
|$\theta_{\text{s}}$ \   | Sun zenith angle\                              |$^\circ$\                      |
|$\theta_{\text{v}}$  \  | Viewing (observer) zenith angle\               |$^\circ$\                      |
|`rAA`                   | Relative azimuth angle between sun and sensor  |$^\circ$                       |
+------------------------+------------------------------------------------+-------------------------------+

: Input variables of the [4Sail]{.smallcaps} model as documented by @feret2022prosail. {#tbl-SAIL tbl-colwidths="[20,70,10]"}

In order to achieve accurate modelling with the [Sail]{.smallcaps} model, specifying the reflectance and transmittance of leaves is crucial.
These values can either be measured directly and integrated into the model or derived from the [Prospect]{.smallcaps} model's predictions.
The coupling of these models was first accomplished by @baret1992modeled following the development of the initial [Prospect]{.smallcaps} model.
This combined model is known as [Prosail]{.smallcaps}, and its latest implementation unifies [4Sail]{.smallcaps} and [Prospect-Pro]{.smallcaps} [@jacquemoud2009prospect].
[Prosail]{.smallcaps} provides an effective means of extending plant pigment retrieval techniques from leaf-level analysis to canopy-level assessment, thereby enriching the scope of vegetation studies.

The [Prosail]{.smallcaps} model simplifies light transport through a canopy into a one-dimensional process, in which light can travel either downwards or upwards.
The canopy is assumed to consist of a homogeneous distribution of leaves.
This simplification imposes certain limitations on the applicability of [Prosail]{.smallcaps} and similar one-dimensional radiative transfer models [@bailey2020one].
Although the assumption proves reasonably effective for modelling canopies that can be approximated as a homogeneous mass, such as grass, wheat, or soybeans, it may not be well-suited for more heterogeneous canopies.
As a result, the [Prosail]{.smallcaps} model is not the most suitable choice for examining crops with intricate canopy structures such as almond trees --- alternative models should be considered and explored for these specific applications.

### Ray tracing

In order to simulate the impact of a specific three-dimensional canopy structure on the reflectance, a technique called ray tracing is required.
Ray tracing is a method for modelling light transport and is widely employed in rendering algorithms to create realistic digital images. It is not the only rendering algorithm, nor is it the fastest, but it tends to deliver the most realistic results [@peddie2019ray].

The underlying reason for this is that ray tracing truthfully replicates the behaviour of light, offering a highly accurate simulation of its interactions within various environments.
Within an environment containing various objects of interest, single rays of light are simulated, each assigned a specified (spectral) reflectance and transmittance.
Each ray of light is represented by a direction and an energy level provided for every simulated spectral band.
When a ray of light encounters an object, its direction changes according to a bidirectional reflectance distribution function (BRDF), and its energy level is reduced by multiplying it with the object's surface reflectance.
The ray of light is continuously losing energy, as an object's reflectance is always $\leq 1$.
Ultimately, the ray either exits the modelled environment or strikes the simulated sensor, contributing to the values of simulated pixels.

Forward path tracing is a technique in which ray tracing originates at the light source and may conclude at the simulated camera.
The primary drawback of this approach is that most light rays leave the modelled environment without impacting the simulated image at all.
Hence, a more computationally efficient strategy begins at the sensor position, sending light rays into the scene while calculating the appropriate spectral intensity for each individual ray.
This process is known as backwards path tracing [@LessUserManual2022].

Ray tracing has advanced significantly over the last few decades, particularly due to the improvement efforts of the video game and computer animation industries.
The implementation of many tricks such as backwards path tracing has constantly improved the performance of ray tracing algorithms.
However, research has primarily focused on methods for increasing algorithm speed [@wyman2021rearchitecting; @lin2022generalized]. Accurate simulation of optical processes has been of lesser importance, as long as the resulting images appear visually pleasing to humans.
Consequently, many ray tracing algorithms developed for computer graphics cannot be directly applied to simulating canopy reflectance [@LessUserManual2022].

### Three-dimensional canopy reflectance models

Specialized programs are necessary to carry out the ray tracing calculations required for scientific applications.
One such program is the Less{.smallcaps} model, specifically designed for simulating forest canopy reflectance.
[Less]{.smallcaps} ([l]{.underline}arge-scale [e]{.underline}mulation [s]{.underline}ystem for forest [s]{.underline}imulation) was developed by @qi2017large in 2017 and has been continuously maintained and updated in subsequent years [@qi2019less; @qi20223d; @qi2023fast].
The model employs a backward path tracing method to generate large-scale spectral images, ranging from the visible to the thermal infrared spectral domain [@LessUserManual2022].

Access to [Less]{.smallcaps} is provided through a freely available graphical user interface (GUI), which facilitates the arrangement of three-dimensional objects within a scene and allows for pre-rendering inspection.
Each component in the modelled scene, such as leaves, stems, or soil, can be assigned specific optical properties --- both reflectance and transmittance --- that influence ray tracing.
However, all simulated surfaces are assumed to be Lambertian, or perfectly matte, meaning that no bidirectional reflectance distribution function can be implemented for leaves in [Less]{.smallcaps} [@qi2017large].

Nevertheless, [Less]{.smallcaps} enables the configuration of numerous parameters, including zenith and azimuth angles for both sensor view and solar irradiance, sensor height, cloudiness type, and the desired quality of the produced image.

It is worth noting that [Less]{.smallcaps} is not the only model capable of rendering a three-dimensional canopy for scientific purposes.
Another example is the ray tracing model [Dart]{.smallcaps} ([d]{.underline}iscrete [a]{.underline}nisotropic [r]{.underline}adiative [t]{.underline}ransfer), introduced by @malenovsky2013retrieval.

## The inversion problem

Thus far, the discussion has centred on radiative transfer models as a method for simulating leaf or canopy reflectance and transmittance, assuming that parameters such as chlorophyll, anthocyanin, or carotenoid content are already known.
This is referred to as the forward mode of a radiative transfer model. In some cases, the forward mode of a model might be useful in and by itself, most notably for generating synthetic data.
Such data can be used for the intercomparison of other models, to conduct a sensitivity analysis, or even to find new vegetation indices [@cheng2014estimating; @jin2014newly].

However, in many cases, reflectance information is already available, and the objective shifts to identifying the set of parameters that, given a specific model, best explain the observed data.
This is referred to as the *backward mode* of a radiative transfer model.
The goal is to discover the inverse function $f^{-1}$ of the actual model $f$.
$$
f: x \to y \qquad \qquad f^{-1}: y\to x
$${#eq-inversion}
Upon successful model inversion, the output of the model becomes its input and vice versa.
Unfortunately, inverting radiative transfer models such as [Prospect]{.smallcaps} is considered mathematically ill-posed and cannot be accomplished algebraically by simply re-writing the equations [@daun2017].
Consequently, alternative methods must be employed to achieve the model inversion.
Specifically, the inversion of a radiative transfer model can be achieved through mathematical optimization, look-up tables, or machine learning techniques.

### Mathematical optimization

The conventional approach to inverting [Prospect-Pro]{.smallcaps} involves mathematical optimization, specifically sequential quadratic programming [@feret2022package; @borchers2022pracma].
This iterative optimization algorithm seeks to identify the parameters $\boldsymbol \theta$ for the model $f$ in order to minimize a cost function $J$.
Typically, the mean squared error is employed as the cost function, which represents the sum of the squared difference between the model's predictions ($f(\boldsymbol {\theta})_\lambda$) and the observed reflectance and transmittance data.
$$
\boldsymbol {\hat \theta}
= \arg \min_{\boldsymbol {\theta}} J(\boldsymbol \theta, \boldsymbol \rho) =
\arg \min_{\boldsymbol {\theta}} \sum_{\lambda=1}^{n} \left( f(\boldsymbol {\theta})_\lambda - \rho_\lambda \right)^2
$${#eq-minimization}
Note that the formula in [equation @eq-minimization] only relates to reflectance but works analogous for transmittance as well.
The optimization algorithm operates by iteratively running the model, numerically approximating the gradient ($\nabla_J$) and the Hessian ($\mathbf H_J$) of the cost function $J$, and utilizing this information to adjust the parameters.
This iteration is stopped when the gradient becomes sufficiently small ($\nabla_J \approx \mathbf 0$).
Inverting the [Prospect-Pro]{.smallcaps} model necessitates approximately 132 model runs on average[^SQP] to achieve a tolerance level of 10^−6^.

[^SQP]: This statement highlights that the average number of model evaluations required for inverting the [Prospect-Pro]{.smallcaps} model on the almond dataset was 132. While this number may vary for different datasets, the differences are not expected to be extreme, remaining within the same order of magnitude.

As an alternative to the maximum likelihood estimation effectively executed via the minimization task in [equation @eq-minimization], a Bayesian framework could be employed to identify the model's parameters [@daun2017].
This method requires information on the prior distribution of the parameter to compute the posterior probability distribution.
Although it is computationally more demanding than the maximum likelihood estimation, it offers the advantage of producing credibility intervals for the parameters based on their joint distribution.
For instance, @zhang2005estimating and more recently @shiklomanov2021cutting employed a Bayesian calibration method to invert the [Prosail-2]{.smallcaps} and [Prospect-5]{.smallcaps} models, respectively. Both teams utilized a Markov Chain Monte Carlo (MCMC) algorithm to achieve this inversion.

There are three primary drawbacks associated with employing mathematical optimization as an inversion technique for radiative transfer models.
The first drawback concerns the feasibility of the approach, which diminishes as the computational cost for a single evaluation of the model becomes too large.
This typically occurs when the radiative transfer model grows increasingly complex and computationally demanding, thereby reducing the efficacy of mathematical optimization as an inversion method.
The second disadvantage emerges when handling a substantial number of observations that necessitate individual parameter retrieval.
In these instances, mathematical optimization loses its effectiveness, as the information obtained from inverting the model for one observation cannot be utilized for another.
The final concern involves the risk of becoming trapped in a local minimum, contingent upon the selection of initial parameters.
If these limitations prove to be insurmountable, it is prudent to explore alternative solutions such as look-up tables or machine learning techniques, which might offer greater efficiency and effectiveness in addressing such challenges.
Consequently, for the inversion of more intricate models like [Less]{.smallcaps}, mathematical optimization is not the most suitable approach.

### Look-up tables

Look-up tables serve as a particularly valuable model inversion technique when the model in question is computationally expensive, yet its results can be cached easily [@selan2005using].
Typically implemented as a fixed-size array, a look-up table consists of elements, each of which corresponds to a specific set of parameters.
In the case of [Prospect-Pro]{.smallcaps}, these elements represent the combined reflectance and transmittance data.
This structure enables relatively rapid data access; however, it also necessitates pre-calculation and storage of the entire look-up table, encompassing all possible parameter combinations [@berger2018evaluation].

The primary disadvantage of look-up tables is the potentially extensive time required for pre-calculating all elements, as well as the necessity of accessing the entire table --- which may represent an exceptionally large data structure --- for making predictions.
It is important to highlight that the data within look-up tables often displays a high degree of collinearity.
As a result, a machine learning model can be trained on the look-up table, significantly reducing its size, and thus facilitating the distribution of the inverted model and also the inference process [@verrelst2016active].

### Model inversion via machine learning

Inverting models through machine learning can be achieved by first generating a synthetic data set using the radiative transfer model, and then training a machine learning model on that synthetic data set.
In this regard, artificial neural networks have gained particular prominence.
As the number of observations rises, neural networks tend to outperform other machine learning models.
Given the synthetic nature of the data, there is virtually no constraint on the number of observations [@kimes2000inversion; @verger2011optimal].
Moreover, artificial neural networks offer the advantage of producing relatively quick predictions during the inference stage once they are trained.
For instance, the canopy level radiative transfer model [Prosail]{.smallcaps} has been successfully inverted for crop trait retrieval using neural networks [@verger2011optimal], but also using random forest regression [@doktor2014extraction].

@verrelst2015replacing refer to machine learning models that approximate the inverse of a radiative transfer model as emulators. The authors argue that the most significant advantages of emulators are their ability to interpolate effectively in regions where the radiative transfer model has not been executed and their rapid prediction capabilities compared to methods like look-up tables.