- In the previous Session we looked at RT modelling of
**leaves** - Now we will consider a
**full canopy** - We will just the basics, as the aim is that you can start using well-established RT models
- Obviously, lots more to cover!

- We will mostly consider a
**turbid medium**canopy - That is a random volume of leaves and air
- In the optical domain, the size of the objects is $>>\lambda$

- Vertical leaf area density function $u_{L}(z)\,\left(m^{2}m^{-3}\right)$,
- Vertical leaf
*number*density function (e.g. the number of particles per unit volume), $N_{v}(z)\,\left(N\,particles\, m^{-3}\right)$

- Distribution of the leaf normal angles, $g_{L}(z, \vec{\Omega}_{L})$ (dimensionless),
- Leaf size distribution, defined as area density to leaf number density and thickness.

The RTE describes the change of **incident radiance intensity** at a specific height and direction $I(z,\vec{\Omega})$.

- Attenuation is governed by Beer's Law $$ I(z)=I(0)\exp(-\kappa_{e}\cdot z)=I(0)\exp(-\kappa_e\cdot LAI) $$
- $\kappa_{e}$ is the product of the medium's
**particle density**and the**extinction x-section**- Also split into radiation
**absorbed**and**scattered away in other directions**.

- Also split into radiation
- Remember that $LAI = \int_{z=0}^{z=-H}u_{L}(z)dz$
**Q***What value of LAI is needed to intercept 99% of the radiation if $\kappa=1$?*

- What if your particles (leaves) are
**oriented**? - We need to
**project**the leaf area density across the beam direction $\vec{\Omega}$ - We project the leaf angle distribution function $u_{L}(z)$ using into $\vec{\Omega}$ by multiplying by

$$
G(\vec{\Omega})=\int_{2\pi} g_{L}(\vec{\Omega}')\left| \vec{\Omega}\cdot\vec{\Omega}'\right| d\vec{\Omega}'.
$$

- From the previous slide, we have that

where $\kappa_{e}=G(\vec{\Omega})/\mu$.

So...

- Attenuation is a function of
- Leaf area
- leaf angle distribution
- direction of propagation

- What happens to $\kappa_e$ when...
- Sun is overhead, vertical leaves?
- Leaves are horizontal?
- Zenith angle is low?
- Zenith angle is high?

- At around $\sim 1 rad$ angle?

- Indicates scattered incoming radiation from all directions into the viewing direction $\vec{\Omega}$.
- $P(\cdot)$ is the
*phase function* - In the optical domain, we tend to specify $P(\cdot)$ as a function of
**leaf area density**and an**area scattering function**$\Gamma(\cdot)$.

$$
\begin{align}
\Gamma\left(\vec{\Omega}'\rightarrow\vec{\Omega}\right) &= \frac{1}{4\pi}\int_{2\pi+}\rho_{L}(\vec{\Omega}', \vec{\Omega})g_{L}(\vec{\Omega_{L}})\left|\vec{\Omega}\cdot\vec{\Omega}_{L} \right|\left|\vec{\Omega}'\cdot\vec{\Omega}_{L} \right|d\vec{\Omega}' \\\\
&+ \frac{1}{4\pi}\int_{2\pi-}\tau_{L}(\vec{\Omega}', \vec{\Omega})g_{L}(\vec{\Omega_{L}})\left|\vec{\Omega}\cdot\vec{\Omega}_{L} \right|\left|\vec{\Omega}'\cdot\vec{\Omega}_{L} \right|d\vec{\Omega}' \\
\end{align}
$$

- $\Gamma$ is a
**double projection**of the leaf angle distribution, modulated by the directional reflectance (upper hemisphere) and transmittance (lower hemisphere) - This is quite similar to $G$
Typically, we assume leaves to be bi-Lambertian, so simplify.... $$ \Gamma\left(\vec{\Omega}'\rightarrow\vec{\Omega}\right) = \rho_{L}\cdot\Gamma^{\uparrow}(\vec{\Omega}, \vec{\Omega}') + \tau_{L}\cdot\Gamma^{\downarrow}(\vec{\Omega}, \vec{\Omega}') $$

Also, if we assume $\rho\sim\tau$ (or a linear function), $\Gamma$ is a weighting of the upper and lower double projections of the leaf angle distribution modulated by the spectral properties of the single scattering albedo.

- Expressions for attenuation and scattering $\Rightarrow$ can solve the RTE.
- Need a bottom boundary (=soil)
- Assume
**only the first interaction**(only one interaction with canopy or soil) - I will skip over the algebra to give an expression for the
**directional reflectance factor**:

- $\exp\left\lbrace -L\cdot\left[ \frac{G(\vec{\Omega}_{s})\mu_{o} + G(\vec{\Omega}_{o})\mu_{s}} {\mu_s\mu_o} \right]\right\rbrace \cdot \rho_{soil}(\vec{\Omega}_{s}, \vec{\Omega}_{o})$
- Radiation travelling through the canopy $\rightarrow$ hitting the soil $\rightarrow$ traversing the canopy upwards
- Double attenuation is given by Beer's Law, and controlled by LAI and leaf angle distribution

- $\frac{\Gamma\left(\vec{\Omega}'\rightarrow\vec{\Omega}\right)}{G(\vec{\Omega}_{s})\mu_{o} + G(\vec{\Omega}_{o})\mu_{s}}\cdot\left\lbrace 1 - \exp\left[ -L\cdot\left( \frac{G(\vec{\Omega}_{s})\mu_{o} + G(\vec{\Omega}_{o})\mu_{s}} {\mu_{s}\mu_{o}} \right)\right]\right\rbrace$
- Volumetric scattering of the canopy
- Controlled by area scattering phase fanction $\rightarrow$ control by single scattering albedo
- Inverse dependency in $G$ and view-illumination angles
- Dependence on LAI too

*When can we ignore the contribution of the soil?*

- Assume a spherical leaf angle distribution function & bi-Lambertian leaves
*What does this mean?*

- If reflectance is assumed to be
**linearly related**to transmittance $ k = 1 + \tau_{L}/\rho_{L}$

- $\cos\gamma=\left|\vec{\Omega}'\cdot\vec{\Omega}\right|$

$$
\Gamma\left(\vec{\Omega}'\rightarrow\vec{\Omega}\right) = \frac{\rho_{L}k}{3\pi}\left[\sin\gamma + \left(\frac{\pi}{k}-\gamma\right)\cos\gamma\right]
$$

- For first order solution, $\rho(\Omega, \Omega')$...
- Refl factor combination of two tersm: uncollided direct & collided volume term
- double attenuated soil return (dependent on leaf angle distribution, LAI, view/illum angles)
- volume scattering: as above, but also dependent on leaf optical properties
- Tends to be larger for larger phase angles

- Remember
*all*the assumptions we made!!! - In the NIR, $\omega$ is quite high, need multiple scattering terms!

- Range of approximate solutions available
- Successive orders of scattering (SOSA)
- 2 & 4 stream approaches etc. etc.
- Monte Carlo ray tracing (MCRT)

- Recent advances using concept of recollision probability, $p$

$$
\begin{align}
\frac{s}{i_0}&=\omega(1-p) + \omega^2(1-p)p + \omega^2(1-p)p^2 + \dots\\\\
&=\omega(1-p)\left[ 1 + \omega p + \omega^2p^2 + \dots\right]\\\\
&=\frac{\omega(1-p)}{1-p\omega}
\end{align}
$$

- $p$ is en eigenvalue of the RT equation $\Rightarrow$ only depends on structure
- We can use this form to describe reflectance if black soil (or dense canopy)
- From Smolander & Sternberg (2005),
$$
p = 0.88 \left[ 1 - \exp(- 0.7 LAI^{0.75}) \right]
$$
- Assuming spherical leaf angle distribution

- The term
$$
\exp\left[-L\frac{G(\vec{\Omega}_s)\mu_o + G(\vec{\Omega}_o)\mu_s}{\mu_s\cdot\mu_o}\right]
$$
is usually called the
*joint gap probability*, and is the probability of a photon traversing the canopy downwards and then upwards without a collision. - We have assumed these two probabilities are independent, which holds in general...

- but what happens if we consider the retroreflection (=backscatter) direction?
- Then, the downward and upward probabilities need to be identical!
- We need a correction factor for the hotspot direction!
- The increased gap probability results in an enhancement of the reflectance factor (the
*hotspot*)

- We have looked at a turbid medium
- This might be acceptable for a grass canopy such as cereals
- But clearly not right for a savanna!

**Clumping**of the canopoy can be encoded as a modulation on LAI, $C$So the gap fraction becomes $$ \exp\left(-L\cdot C\frac{G(\Omega)}{\mu}\right) $$

We can think of the LAI of a clumped canopy as being

**effective**if $C\neq1$- Canopy types:
- Random distribution: For each layer of leaves, there is 37% overlapping. $C=1$.
- Clumped distribution: For each layer ofleaves, there is more than 37% overlapping. $C < 1$
- regular distribution: For each layer of leaves,there is less than 37% overlapping. $C > 1$

Clumping has an effect on the radiation regime inside the canopy

$\Rightarrow$ effect on GPP

- In the plot
- Case I: LAI and clumping considered
- Case II: clumping considered
- Case III: effective LAI

- Case II vs Case I: ignoring $C$ results in an overestimation of sunlit leaves $\Rightarrow$ increase in GPP
- Case III vs Case I: underestimation of shaded leaf LAI $\Rightarrow$ decrease in GPP

- Need to deal with the mutual shadowing of e.g. tree crowns, soil, etc. $\Rightarrow$ Geometrical optics (GO)
- A first stage is to assume that crowns are turbid mediums
- Calculate scattering & attenuation
- Deal with shadowing

- Hard to measure many processes (e.g. contribution of multiple scattering)
- $\Rightarrow$ use simpler models as a
*surrogate reality* - $\Rightarrow$ aim here is to understand e.g. effects of assumptions, etc
- RAdiative transfer Model Intercomparison (RAMI)

Summary

- Turbid medium approximation
- 1st O: attenuation & volume scattering
- leaf angle distribution, geometry, LAI...
- Assumptions to make modelling tractable

- Multiple scattering
- $p$-theory

- Hotspot effect
- Discontinuous canopies
- RAMI-type efforts