Measuring vegetation is not easy. Estimating would be a better word. But measuring/estimating vegetation is extremely important, so it is worth the effort. Specifically, we are interested in the theory behind a device called ceptometer. We recently bought an ACCUPAR LP-80 PAR/LAI from METER group and the idea is to take advantage of it as much as possible. They have a really good list of resources, especially this guide.

A first important concept is that of PAR, which stands for Photosynthetically Active Radiation. It is the solar spectral range for which Earth beings with photoshynthetic capabilities are able to perform its function. Its wavelength range is [400,700] nm. (Other beings can use Sun's light beyond this range, for example in the near-infrared.) This matches, more or less, the visible spectrum, which is about [380,750] nm. Photons more energetic than that and the cells can be damaged. Less energetic than that and photosynthesis cannot be produced.

Chlorophyll basically absorbs red and blue, which is the reason plants appear green, since part of it is reflected back to our eyes. But not all green light is reflected: part of it is transmitted to deeper layers of leafs and produces photosynthesis as well. Part of this transmitted light is even able to go through the entire leaf, reaching plants below the canopy (region of plant crowns, typically of trees) and helping them grow.

It is clear now why measuring PAR is so important. It allows to measure the amount of vegetation above your sensor if you are able to measure above as well. But it also tells you if a place is viable for agriculture or not. The LP-80 measures PAR in units of micromoles per square meter per second (μmol/m²/s), which technically is the Photosynthetic Photon Flux Density (PPFD). Basically, it quantifies the amount of light energy available for plants to perform photosynthesis. The irradiance (radiant flux received by a surface per unit area) of PAR would be measured in W/m². It seems weird to use mols instead of irradiance, but there is a reason for that: photosynthesis is a quantum phenomenon, and its rate is proportional to the number of photons, not to their total energy.

It is interesting to compare PAR with other quantities, like luminous flux (measured in lumens, lm) or illuminance (measured in lux, lx = lm / m²). These two quantities are human-centric, based on human perception of light, which has a strong bias towards overvalue green.

An interesting quantity here is the production of dry matter P by a plant canopy. It is the process by which plants accumulate organic matter in their tissues, excluding water content. It is a crucial indicator of plant growth and productivity. We can assume that P is proportional to three quantities: 1) S = the flux density of incident radiation potentially intercepted by the plant. In other words, the PAR measured above the canopy with the sensor face upward; 2) f = fraction of absorbed light, i.e. the proportion of incident photosynthetically active radiation that is actually absorbed by the plant canopy (dimensionless); and 3) e = light use efficiency or quantum efficiency or conversion efficiency, representing how effectively plants convert light into biomass during photosynthesis. Typically expressed as grams of dry matter produced per unit of absorbed light energy (e.g., grams per megajoule of PAR). Our formula needs to express this story: some radiation arrives to the leaf (S), then part of it is absorbed (f), and then part of it is converted to dry matter (e):

P=S·f·e.

Out of these three quantities, only S is an environmental one. The other two depend more on the crop and its management. Usually, e is determined by measuring P, S and f. When genetically modifying a crop, the effect on e and f is assessed.

Incident radiation can either be reflected at, absorbed by and transmitted through the canopy. The transmitted fraction can then either be absorbed by the ground or reflected by it, a part that incidentally interacts with the canopy again. A first and naïve radiative equilibrium equation can be (positive is what enters the canopy, and negative what leaves it). The total balance is what is intercepted:

intercepted = incident - reflected - transmitted

If we want to be more precise, then:

intercepted = incident - reflected - transmitted + rebounced from ground

If dividing everything by incident (total), then,

f = 1 - r - t - t·r_s,

where r is the reflected (up) fraction above the canopy, f is the absorbed fraction, t is the transmitted fraction and r_s is the fraction of the transmitted radiation that is bounced back (the reflectance of the soil), so that t·r_s is a radiation that escapes along with r.

The majority of the incident radiation (within our PAR range) is absorbed by the canopy. Then, we can just write the fractional interception f as:

f ≈ 1 - t.

But we can measure f without this approximation. As said before, S is the PAR when the sensor is above the canopy facing upward. We define R as the PAR above the canopy but facing downward, so it catches the reflected radiation. We define T as the PAR below the canopy facing upward, so it catches radiation transmitted through the canopy. Finally, we define U as the PAR below the canopy but facing downward, so it catches the radiation reflected from the soil. We can calculate now our previous fractions as:

t = T / S

r = R / S

r_s = U / T

With t, r and r_s, we can calculate f without the aforementioned approximation.

In order to calculate the LAI (L), there are different approximations. One of them is considerably accurate yet it retains mathematical simplicity, so that is what the LP-80 gives:

L = ( (1-0.5/K)fb - 1)·lnτ / (A (1-0.47fb) ),

where we need to define some quantities:

K is the extinction coefficient for the canopy (the attenuation of a radiation as it travels through the canopy), which can be estimated from the zenith angle Θ (the angular elevation of the sun in the sky with respect to the zenith), with the equation K = 0.5/cosΘ. This formula is valid when the leaf angle distribution (χ) is 1, meaning the orientation of leaves is considered spherical. If horizontal leaves predominate, then Χ would be greather than 1, and for vertical leaves, like grass, less than 1. The value of χ is something to be estimated and given as input to the ceptometer. Otherwise it is assumed to be 1.

fb is the beam fraction: It is the ratio the beam radiation of the sun and the diffuse radiation from sources like water vapor, aerosols, clouds... The ceptometer calculates fb by comparing incident PAR with the solar constant, the value we would expect from a clear day. A value fb = 0 indicates an overcast day. A very cloudy, hazy day needs a value close to 1.

τ is the ratio of transmitted and incident PAR: This ratio is calculated measuring incident radiation above the canopy (sensor facing up, button UP) and measuring transmitted PAR near ground below the canopy (sensor up, button DOWN). A low value of LAI means most incident radiation is transmitted. This means τ close to 1.

A is a parameter calculated as A=0.283+0.875a-0.159a², where a is the leaf absorptivity in the PAR range, assumed as 0.9.

In summary, A is taken as constant, fb is estimated comparing incident PAR with solar constant and K is taken from the geographical coordinates. This leaves τ as the only real quantity to be measured.

There is a question that bugs me: vertical green walls. How to use the ceptometer in this case? here it is recommended to measure outside and inside the wall during an overcast day, to avoid issues with the Sun's angle. But what if we have a sunny day? I would suggest the following: if the Sun does not directly hit the wall, then any orientation is right, since fb will be 0. If the Sun hits the wall, then we can orient the sensor towards the Sun, both outside and inside vegetation. The ceptometer will give a LAI that is not useful, but it also gives τ. We can manually recalculate LAI considering that Θ, the zenith angle, is 0°. Finally, whatever LAI is given, we multiply the value by sinΘ, where Θ would be the zenith angle. With the previous method, without the sine yet, we are considering the vegetation along a diagonal, so the thickness of vegetation is interpreted as this diagonal. Since we can consider the LAI to be proportional to thickness, we correct it with sinβ. This method would not work with the Sun being really high in the sky. And the measurements need to be in such that, if the vegetation has thickness d and height, the sensor cannot be above (h - d/tanΘ) when measuring inside. Another choice could be leaving the sensor facing upward and then just multiplying the LAI from the ceptometer by sinΘ.

A crucial point here is that not only leaves block radiation. As we can read here, ``the indirect methods do not measure leaf area index, as all canopy elements intercepting radiation are included. Therefore, the terms of plant area index (PAI) or surface area index (SAI) are preferred if no correction to remove branches and stems is made.'' So other methods should be explored as well.

The LAI can also be measured with destructive techniques. One of them is to take the harvested leaves from a region, scan them and analyse the images with software. Could we do this to estimate the LAI in a clever, heuristic way? In the wonderful book *Geometry for Entertaintment* by Yakov Perelman, he proposes to measure the area of leaves by considering they are similar to each other, for the same plant. Then, if we can measure one leaf's area with precision and can have an idea of an equivalent length in others, we could, in principle, estimate the LAI. In particular, if we take the length of the leaf at this quantity, we will work with what is known as Thompson's model. How to achieve that? We could consider a region of vegetation. Not a big one, so that we can count, more or less, the number of leaves there. Then, we can decide which *linear* feature to measure, and measure it for these leaves in the region. Then, assuming that their area ratios with respect to our sacrificed leave is proportional to the square of the length ratios, and assuming a distribution of lengths from our measurements, we could give an estimation of the LAI.

Say, for example, that we choose a region of 50×50 cm² of soil, where we count 17 leaves, with a linear feature of lengths 2,3,5,7,2,6,9,12,2,4,6,7,3,4,9,11,3. From a close region, we take a leave and measure its area as 33 cm² (we can flatten it a bit against a paper and draw its contour, that's all), and its linear feature is 7 cm. Then, we can consider that the total leaf area in our region is

33·[ 3·(2/7)² + 3·(3/7)² + 2·(4/7)² + (5/7)² + 2·(6/7)² + 2·(7/7)² + 2·(9/7)² + (11/7)² + (12/7)² ] ≈ 466.714 cm².

This means we have 466.714 cm² of leaf over 50×50=2500 cm² of soil. Then, for 10000 cm² = 1 m² of soil we would have 1866.857 cm² of leaf area. This means that the LAI is approximately 1.87 or 1.9 or even 2.0. A reasonable value.

If the plant has a hierarchical structure, with many leaves grouped in chunks, than we can apply the same philosophy: to measure a chunk with detail, and then to measure a linear feature of it to other chunks. Say we see a region where the plants form clear chunks that are similar to each other. There are 8 of them in that region, and with ratios 4,3,1,2,3,4,5,1 to 1 with respect the linear feature of a chosen chunk. Then, each chunk is composed of 6 similar sub-chunks. For our chosen one, we see that a linear feature of them is in ratios 4,7,2,5,7,1 to 1 with respect to one of them (the last one, of course). Then, each sub-chunk as 7 leaves, with linear rations 1,1,3,2,2,1,2 with respect to one of them.

The LAI could be calculated as follows. We begin with the small scale and measure the precise area of a leaf. We get 12 cm². Now, we know that that sub-chunk will have an area of 1·(1²+1²+3²+2²+2²+1²+2²) = 288 cm². Now, all the sub-chunks will add to 288·(4²+7²+2²+5²+7²+1²) = 41472 cm². Finally, all the chunks add to 41472·(4²+3²+1²+2²+3²+4²+5²+1²) = 3359232 cm² = 335.9232 m². If the region had a soil base of 23 m², then the LAI would be 14.6. I have no idea if this method would bring reasonable results, but it makes sense to me. And, if reasonable, quite a big region could be measured with a very reduced set of measurements. Self-similarity between leaves and hierarchical structures can really facilitate the task.

Unfortunately, it seems leaves are not similar. As we can learn here, with pdf here, ``leaf area–length data of some plants have been demonstrated not to follow the principle of similarity''. But not all is lost. Instead of using an exponent equal to 2, perhaps we could use a different value. Such quantity could be called the allometric exponent. In this field, it is common to study area-length allometry (very common) and leaf weight-area allometry (not so common). But we are interested in the latter.

Montgomery came up with this formula: Area = c·(leaf length)·(leaf width) = c·l·w, where c is just a parameter to be adjusted. The length is defined as the distance from leaf apex to leaf base, and the leaf width is defined as the maximum length of the segments perpendicular to the straight line passing through leaf apex and leaf base. The authors of this article find that this formula is really good for several types of tree leaves. No exponents are required, just a single parameter. However, this parameter changes with the type of leaf, ranging from 0.5 (triangular leaf) to π/4 ≈ 0.785 (elliptical leaf with major axis as length). This is more involved, of course, and it tells nothing about the allometric relations of higher structures. This article is from 2019, so there is a lot of interesting work to be done in this field.

Disclaimer: This text is just a collection of written notes resulting from internet browsing (mostly Wikipedia, and I am a proud donor) and the LP-80 manual. Thus, no self-credit is claimed here, except the proposal for vertical measurements and the discussion of heuristic measurements.