Light Absorption by a Thick Layer
Edward B. Lochocki
Source:vignettes/web_only/thick_layer_absorption.Rmd
thick_layer_absorption.Rmd1 Overview
This vignette explains my understanding of the formula used in the
thick_layer_absorption function found in the C++ source code file
src/module_library/sunML.cpp. Although it has been used in BioCro, this
equation seems to be rarely discussed in the plant biology literature. As far as
I can tell, it is ultimately based on Equation (1) from
Saeki (1960). However, the description in that paper is short
and very sparse regarding details. Here I have tried to add some explanatory
comments and fill in a few math steps that were skipped. I have tried to follow
the “spirit” of that paper, although I have made a few notational changes. See
Section 4 for more information about tracking this equation
through various references.
2 Absorption by a Generic Material
Suppose light of intensity \(I_0\) (representing a flux density of photons or energy, expressed in units of photons per area per time, or energy per area per time) is incident on the perfectly flat surface of an infinitely thick piece of material that reflects, absorbs, and transmits light. Assume that the optical properties and density of the material are constant along any plane parallel to the material’s surface, although they may vary with depth within the material. In this situation, the light intensity at any point within the material will only depend on the distance from the material’s surface, so we can reduce the number of relevant dimensions from three to one, an important simplification.
As the light passes through the material, its intensity will gradually diminish until it eventually reaches zero. We can express this mathematically via a one-dimensional expression \[\begin{equation} I(x) = I_0 \cdot f(x), \end{equation}\] where \(x\) is a coordinate that represents the amount of material the light has passed through, and \(f(x)\) is the fraction of the original light received by the material at \(x\). Note that if the material does not have uniform density, \(x\) is not a spatial coordinate, but instead will have a non-linear dependence on distance. Although we do not know the particular form of \(f(x)\), we can safely assume that:
\(f\) is monotonic.
\(f(0) = 1\) (so the incident light at the material’s surface is \(I_0\)).
\(f(x)\) approaches 0 as \(x\) approaches infinity (so the light intensity is fully diminished deep within the material).
Now consider two points \(x\) and \(x + \Delta x\) within the material, separated by an amount of material \(\Delta x\). The change in light intensity per unit material between these two points, \(\Delta I/\Delta x\), can be expressed as \[\begin{equation} \frac{\Delta I}{\Delta x} = \frac{I(x + \Delta x) - I(x)}{\Delta x}. \end{equation}\] Rewriting this using \(f(x)\), we have \[\begin{equation} \frac{\Delta I}{\Delta x} = I_0 \cdot \frac{f(x + \Delta x) - f(x)}{\Delta x}. \end{equation}\] If we assume that the change in intensity is due to the absorption and reflection of light as we pass from point \(x\) to point \(x + \Delta x\), we can also write \[\begin{equation} \frac{\Delta I}{\Delta x} \approx - I(x) \cdot \left( A(x) + R(x) \right), \end{equation}\] where \(A(x)\) and \(R(x)\) are the fractions of incident light absorbed and reflected by a thin layer of the material at \(x\) and the negative sign indicates that the overall light intensity decreases due to absorption and reflection. Thus, we can equate the two expressions for \(\Delta I / \Delta x\) to find that \[\begin{equation} I_0 \cdot \frac{f(x + \Delta x) - f(x)}{\Delta x} \approx - I_0 \cdot f(x) \cdot \left( A(x) + R(x) \right) \end{equation}\] or, equivalently, \[\begin{equation} \frac{f(x + \Delta x) - f(x)}{\Delta x} \approx - f(x) \cdot \left( A(x) + R(x) \right) \end{equation}\] As \(\Delta x\) approaches 0, the approximation becomes exact and we can recognize the left-hand side of this equation as the derivative of \(f(x)\) with respect to \(x\): \[\begin{equation} \frac{df}{dx}(x) = - f(x) \cdot \left( A(x) + R(x) \right). \end{equation}\] Rearranging, we can express \(f(x)\) in terms of its derivative and the material’s optical properties: \[\begin{equation} f(x) = - \frac{\frac{df}{dx}(x)}{A(x) + R(x)}. \end{equation}\] Noting that all light received by a thin layer of the material must be reflected, absorbed, or transmitted, we can use \(A(x) + R(x) + T(x) = 1\) to rewrite this equation in terms of the fraction of transmitted light \(T(x)\): \[\begin{equation} f(x) = - \frac{\frac{df}{dx}(x)}{1 - T(x)} \tag{2.1} \end{equation}\]
2.1 Additional Considerations (Especially for Plant Biology)
This simple model for calculating light intensities within a material was first applied in the context of plant biology by Monsi and Saeki (1953), where it is used to calculate light levels within a plant canopy. This application is the main reason for why we describe \(x\) as an amount of material rather than a depth; in plant canopies, the leaves are not uniformly distributed with depth, so the cumulative leaf area index is a better independent variable for light absorption.
Monsi and Saeki (1953) is difficult to find and was written in German, but parts of it have been reproduced in English and are easier to access (Saeki 1960, 1963; Hirose 2004). Saeki (1960) notes the following about this equation:
It must be noted that in these equations \(m\) includes not only the fraction resulting from light transmitted through the leaf blades but also the fraction reflected downward from inclined leaves. This \(m\) is not constant but increases with the depth of foliage, because light of particular wavelengths is more liable to be reflected and transmitted, and increases in proportion at deeper positions.
(In the original notation, \(m\) was used in place of the \(T(x)\) we use here.) Thus, \(T\), \(R\), and \(A\) are not exactly the same as the corresponding optical properties of an isolated layer of the material (such as a leaf).
3 Total Absorption
If we consider light from a sufficiently narrow wavelength band, then it may be reasonable to suppose that \(T\), \(R\), and \(A\) are constant throughout the material. In this case, it is possible to estimate the total amount of light absorbed by the material. To do this, we first calculate the absorbed light at depth \(x\) per unit material (\(d I_\text{abs}(x) / dx\)), using \(d I_\text{abs}(x) / dx = I(x) \cdot A\). Substituting in Equation (2.1) and recalling that \(A = 1 - T - R\), we have \[\begin{equation} \frac{d I_\text{abs}}{dx}(x) = - I_0 \cdot \frac{1 - R - T}{1 - T} \cdot \frac{df}{dx}(x). \end{equation}\] Now we can integrate this across the entire range of \(x\) (0 to \(\infty\)) to find the total amount of light absorbed by the material (\(I_\text{abs (total)}\)): \[\begin{equation} I_\text{abs (total)} = I_0 \cdot \frac{1 - R - T}{1 - T} \cdot (f(0) - f(\infty)). \end{equation}\] By assumption, \(f(0) = 1\) and \(f(\infty) = 0\), so this evaluates to \[\begin{equation} I_\text{abs (total)} = I_0 \cdot \frac{1 - R - T}{1 - T} \tag{3.1} \end{equation}\]
Note that Equation (3.1) agrees with intuition in two extreme situations:
If the material does not reflect any light (\(R = 0\)), then Equation (3.1) reduces to \(I_\text{abs (total)} = I_0\). This makes sense because even if thin layers of the material transmit light, there is no way for any light to avoid being absorbed by an infinitely thick layer if there is no reflection.
The other situation is where the material does not transmit any light (\(T = 0\)). In this case, Equation (3.1) reduces to \(I_\text{abs (total)} = I_0 \cdot (1 - R)\). This makes sense because the optical properties of a material with no transmission would be determined only by its surface; the surface would have the same optical properties as any thin layer, reflecting a fraction \(R\) of the light and absorbing the rest.
3.1 Additional Considerations (Especially for Plant Biology)
Although Equation (2.1) was originally developed for plant canopies, it does not rely on any specific properties of canopies and can in principle apply to any material. (In fact, we have written this derivation in a material-agnostic way to emphasize this.) Thus, Equation (3.1) can also apply to a wide variety of materials. The absorption and reflection of light by soil is another situation where Equation (3.1) may be useful, as the assumption of a thick layer that does not transmit any light through it is certainly justified in that scenario.
Because it assumes a thick layer of a homogeneous light-absorbing material, Equation (3.1) is not appropriate for use in a layered canopy model or one that makes distinctions between different leaf classes (such as sunlit and shaded). It is best used for situations such as estimating whole-canopy transpiration or soil evaporation, where it is useful to know the total solar energy absorbed by a thick layer of leaves or soil. Care must be taken even in this case, however, since this equation would still not be appropriate for the small canopies of young plants, which certainly transmit a significant fraction of the incident light.
4 Caveats From the Author
Although Equation (2.1) can be found in Monsi and Saeki (1953), Saeki (1960), and Saeki (1963), Equation (3.1) is not included in those references. So, although this derivation makes sense to me, there is still a chance that it might not be correct. Equation (3.1) can be found in Humphries (2002), the WIMOVAC code, and the BioCro code. In these places, it is variously attributed to John H. M. Thornley and Johnson (1990) or Monteith and Unsworth (1990). Unfortunately, these textbooks are not available in electronic form. I have looked in a library copy of Monteith and Unsworth (1990) and in electronic versions of newer editions, but have not been able to find this equation. I have attempted to find an explanation for Equation (3.1) elsewhere but have not been successful so far. J. H. M. Thornley (2002) discusses Equation (2.1), but ultimately just references the (currently inaccessible) textbook John H. M. Thornley and Johnson (1990).