Canopy Light Extinction (Beer's Law for a Canopy)

This is Beer–Lambert. Not an analogue of it, not something like it — the same equation, moved from a cuvette to a forest. A chemist writes $A = \varepsilon b c$ and reads absorbance; Monsi and Saeki in 1953 wrote $I = I_0 e^{-kL}$ and read light under a canopy. Leaf area index plays the part of concentration times path length, and the extinction coefficient $k$ plays the part of molar absorptivity. Convert base 10 to base e and the two are identical.

It does not stop there. A radionuclide decaying, $N = N_0 e^{-\lambda t}$; chlorine disappearing down a water main, $C = C_0 e^{-kt}$; a capacitor discharging through a resistor, $V = V_0 e^{-t/RC}$; litter rotting on a forest floor, $X = X_0 e^{-kt}$. Six equations, six trades, six notations, one relation. Every one of them is the solution to the same differential statement: the rate of loss is proportional to how much is left. When that is true, this is the answer, and it does not matter whether the thing being lost is photons, atoms, chlorine, charge or leaf litter. Every one of them has a $1/k$ that is a characteristic scale — a mean free path, a mean lifetime, a time constant, a turnover time. Someone who learned the RC time constant in a wiring course already understands litter turnover and has not been told.

What is specific to plants is the size of $k$, because k is a leaf-angle number. A canopy of steeply held leaves — ryegrass, wheat, most conifer shoots — presents little area to an overhead sun and runs $k \approx 0.3$ to $0.5$. A canopy of flat, horizontally held leaves — most broadleaved trees, clover, a planophile soybean — intercepts nearly everything it can and runs $k \approx 0.7$ to $1.0$. A random, spherical leaf-angle distribution against a vertical sun gives $k \approx 0.5$. That spread is the whole reason an erect-leaved crop can carry LAI 8 profitably and a horizontal-leaved one cannot: the erect canopy rations the light down through many layers instead of burning it on the first.

Three limits ride along. $k$ is not constant through the day — a low sun sees more of an erect leaf, so $k$ rises toward morning and evening, and a noon value understates daily interception. The law assumes leaves are randomly scattered in a horizontally uniform layer, which real canopies are not; clumping lets more light past than this predicts, and the usual repair is a clumping index multiplying $L$. And the equation says nothing about wavelength: $k$ for photosynthetically active radiation is far higher than $k$ for near-infrared, which leaves reflect and transmit freely. That difference is not a nuisance — it is precisely the signal NDVI is built on.

Measured properly, $k$ is a regression rather than a division: read transmitted light at several heights, plot $\ln(I_0/I)$ against the cumulative leaf area above each sensor, and take the slope through the origin. Do it under uniform overcast if you can, because diffuse light removes the sun-angle dependence and returns a $k$ that belongs to the canopy rather than to the hour.