In fact there is no difference to other plot designs, you can do exactly the same also in fixed area or nested plots... If you calculate the expansion factor per tree and expand e.g. tree basal area to one hectar, then you can sum this over all of your trees (from multiple plots) and divide by n. Same result! Sum(y_i) (plot aggregate) is here equal to Sum(y_ij) (sum over trees). Only that you need no expansion factor here, since you are already counting on a per ha basis

I am confused: this is only for counting factor k=1? For measurement in meters and cm the ratios for k=1,2,4 are 1:50, 1:34.5 and 1:25 respectively (both in same units). Means, using k=1 (every counted tree is 1m²/ha) a tree with dbh of 35cm (0.35m) has a maximum distance of 0.35*50=17.5 meters SORRY, just realized that k is not counting factor (as used here in Germany). Our "k" is your BAF and what you notate as k is what we call c...

very right! It is usually doubtless for many trees that are definitely IN, for those close to the border it is difficult to guess and a distance measurement is indicated...

No, its proportional to its basal area (it is the radius of a circle that is proportional to dbh, then the circle area is proportional to dbh^2 or basal area)

Yes, e.g. our new Dendrometer III: https://doi.org/10.25625/USXNYO

probability is proportional to basal area (while plot radius is proportional to dbh)!

If you leave out the 3*F it becomes more general. Trees per acre is just the sum of tree expansion factors. Then it can also be used for unequal probability designs in which expansion factors might vary with tree size

density?

There are also statistical arguments: systematic sampling is known to be more precise.

Ok, finally here it comes, thank you! This is quite late and maybe an explanation of different population concepts would be nice at an earlier stage already. Harvard Forest was a special case (rather uncommon in inventories but maybe meaningful inside this experimantal plot) and now finally you introduce infinite population. From my own experience: we have sometimes similar problems here, since we teach general sampling statistics first (with fpc) and then switch to infinite.

No mentioning of Bitterlich in this paragraph?

Only for fixed area plots (rather uncommon). For nested plots or Bitterlich this is not he case

which is in fact also plot sampling, only with plot size proportional to basal area

"for each location" would only hold for fixed area plots. As soon as a unequal probability design is used (nested plots, Bitterlich), we would expand on the tree level and aggregate to plot level afterwards.

or systematic

or just "sampling locations"

Sorry, but this is completely wrong! We have full control over the population variance (except practical constrains). If the population consist of plots, then we are the ones who define the character of this population! Imagine you make your n=100 plots twice as large as the area of interest (means: 100 times full census), then the population variance among these plots is zero! Any plot size smaller than the area of interest will introduce variability among the observations. And in your case of subdividing the area into cells: Of you make them larger, the variance among them will get smaller...

And now assume you would allow such a quadratic plot at any location, then is getting infinite. BUT: contrary to finite population in which we expect a different response in each cell, this does not hold in an infinite population. Many sampling locations (also infinite) would lead to the same included trees. See https://youtu.be/-8CVcXOKRxM?si=MpWdojUe3FbcU-Mn

the estimated population variance

only in finite population

which is the standard case if we consider an infinite population ;-)

error variance (or standard error). Should be distinguished from population variance here!

maybe "sampling element" is better

Ok, somehow you like to stick to this finite definition of the population. In the next sentence, however, you say (correctly) that each possible sample must have a non-zero probability. In real life: you could select a sample plot at any point in the total forest area. In a finite population view this is not the case (and somehow you are limited to quadratic cells as plots, which is not in line with what you explain later)?? Why is it so important to define the population as finite here? Imagine we conduct SRS by random selection of x,y coordinates in the area (and install plots there). Then we are outside of your population concept (the population of possible plots is infinite).

... or the total forest area

Which means you need to consider finite population correction in many estimators. Why not infinite populations (which is what we are doing in reality)? The concepts you introduce later (SRS, stratified, double sampling, ... ) would hold in exactly the same way, only that you can ignore the fpc

Which would be a very untypical strategy, right? We see such examples in old textbooks (like Shiver and Borders and my former boss Akca used it too). Such a subdivision of the area would result in a finite population of cells. No other overlapping cell in between would have a positive probability to be selected. If you like to avoid this, you can just say that a sample of plots is selected from the total forest area.

If the population we are looking at does allow, yes. We can e.g. measure all trees if we are interested in tree characteristics. It becomes impossible if the population consists of plots (resp. all possible sample plots in an area).

It is not at all a bad idea to introduce dplyr here. data.table would be an alternative and often more efficient on large datasets, but the code is not as readyble

If these are standard, fine. Since this is just to learn for the students, you can also use simpler formulations like Smalian.

instead of looping, we would rather use lapply here later

For a simple example ok, but later you want to have a leftjoin here and read the model coefficients for different species from a different related table. But you can do that also in base R:

seperate table: model coefficients by species

coef_df <- data.frame( species = c(your species 1, 2,3,...), a = c(0.5, 0.7, 0.6), # coefficients b = c(1.2, 1.5, 1.3) )

join coefficients on trees

library(dplyr)

trees_with_coef <- trees_df %>% left_join(coef_df, by = "species")

But dplyr was not yet introduced and i understand that they first should do it by hand...

This is the log transformed linear form that we often use for regression analysis (due to heteroskedasticity in the metric data). As you see the coefficients are here ß and should be estimates by a linear regression. However, if it comes to applying an allometric model we usually use the form AGB=a*DBH^b. ß1 is b and a is exp(ß0). I would recommend to substitute ß0 and ß1 by a and b and use the above formulation. Easier to digest for the students and more common in practice.

I would not recommend to introduce any "foresters constants", it is just simple geometry if we want to calculate the area of a circle from its diameter. Students tend to learn such things by heart and forget about the fundamentals... Here the pi/10.000 comes in because DBH comes in centimeter (conversion from cm to meter) and the strange (DBH/2)^2 substitutes pi/4. I prefer to explain my students how to calculate the area of a circle and remind them that we want to have this in meter, instead of confusing them.

you can have the same basal area with a single big tree or hundreds of small trees. There is not necessarily a relation between BA and density

closely related

Maybe good to start with a single skript, but usually later you want to have a project with multiple scripts in which you can separate certain things from each other (e.g. functions and models from calculations or data import). Just see that this comes in 3.4!

Maybe shift this chapter up and describe the datasets after?

Mhhh, here you maybe introduce some confusion for the reader. It is correct that in this specific experimental plot and for the specific ecological perspective the interest is on trees. And yes, if the interest is on tree characteristics, the total population is all the trees. But this is very different to forest inventories (observational studies) that aim at describing the fores area. There, the population is the forest area and the sampling units are subsets of this area (plots). Maybe substitute by "... is a full census or complete enumeration of all trees, which in this case represent the population"

Somehow difficult to consider printed and online at the same time. Some of it might be shifted to the caption.

...and plot measurements instead of point mesuerements

ok, you here mention sample plots for the first time

usually "canopy height model CHM". Which is a surface...

Well, this is what everybody says, but if we look at what is used for decision making at the end, I am not sure. Maps are great for communication, but calculations are done in tabular data. If we provide forest managers with high resolution wall-to-wall maps, they usually ask for simpler products showing useful classes. My experience is that we often aggregate our high resolution back to something more simple. In the carbon business emission factors are not provided at the pixel level, but e.g. for a certain forest type.

I think wood chemistry suggest rather 0.47, but I have no reference at hand

Maybe substitute "LiDAR based mapping" by "LiDAR-based or -assited estimation". "Mapping" is confusing here

In the meanwhile LiDAR data acquisition has emerged to a standard in many countries. The wording "calibration data" reflects a purely model based perspective (common in remote sensing), while we usually look at it as ancillary data

Just comment: Here we look at a typical experimental study that allows hypotheses testing and significance (in contrast to the observational studies mentioned above in which we can maybe look at relationships but never on cause and effect). Such studies do not aim at describing a forest area, but are designed to research into ecological or other relationships. Good to have such different examples!

I stop commenting on point or plot here, since you consistently used point. At the end it is a question of personal style and "point" is also fine as long as there is a mentioning that trees around such a point are included based on a certain rule (and I assume you come to plot designs later). Speaking of points may help to explain the infinite character of our population later.

sample plot

Considering a later printed book publication you might want to shift this into the figure caption

Sampling locations were selected in a systematic grid

sample plots? Maybe you come to observation units in a later chapter, but i see no reason why not to mention it already here. Tree characteristics cannot be measured at a point

Alternatively: Selected plot locations

or "selection of measurement units". The typical target variables cannot be observed at a dimensionless point (as you say), usually our sampling elements are small outcuts of the forest area.

..., which is a model prediction and in fact the mean of observed values at this dbh

here: "from a statistical model". If it would be an allometric model of the form BM=a*dbh^b it would look different

see, we are not looking at strict allometry here (which would be a process model perspective) but look for the best fit. In this case dangerous if you interpolate beyond the range of data. Look at Betula lenta where the best fit to the current data would suggest that biomass is decreasing with increasing dbh. In regard to model choice, a data analyst has both in mind: a possibly good fit to the current data and biological plausibility.

Instead of "create" maybe use "fit a model using regression techniques" or something

"Allometry" or allometric relations usually refer to a specific kind of relations (the relation between two relative growth rates in one individual). We tend to call many model "allometric" that are in fact other kinds of relationships. Taking the character of allometric relations serious would mean to apply a power model, but in fact we are often using others. Anyway, I would not change the text here because it is in line with the general understanding.

If at all it would be very destructive

The target variables we are typically interested in are rarely "measurable" (volume, biomass, biodiversity, ....). It is another important task of the data analyst to to identify and calculate the essential target variables or requested information from the data (features) at hand. In most cases this require the application of models.

It is maybe helpful for the reader to distinguish between typical observational studies (forest inventories aim at estimating status and change) and experimental studies (investigating effects of treatments or researching into dynamics)

Maybe include Forest Inventory - Methodology and Applications by Annika Kangas and Matti Maltamo and Sampling Methods, Remote Sensing and GIS Multiresource Forest Inventory by Köhl and Steen Magnussen.

I can confirm ;-)

From a certain plot size onwards, the gain in precision is minor and increasing sample size instead of enlarging plots is statistically more beneficial. For typical variables like volume, empirical research shows that 15-20 trees per plot are enough. But there are other target variables that might justify larger plots...

Beautiful figures!

If horizontal distance is measured no slope correction is required...

"point sampling" is just a form of "continuously nested plots" and not at all "plotless". There is no argument to look at it in a different way.

Not consistent with the infinite population view described above!! The population is the infinite number of possible plot locations (continuum of sampling locations), from which some are selected. In some textbooks you find examples of partitioning the area into a finite number of grid cells from which some are selected. Then FPC is required, but this view is not consistent with what we usually do in forest inventories...

Infinite small! We draw a sample of plots from an infinite number of available plot locations in the area. The sample is therefore not shrinking the remaining population size in the next draw...

This is not correct! FPC applies to any sampling design if applied in a finite population. It is a correction that ensures that the standard error becomes 0 if all elements of the population are in the sample. It is relevant if the sample proportion is relatively large (and the remaining u sampled portion becomes small). The argument for not considering it in forest inventories is that we sample from an infinite population...

For sampling of single fixed area plots... Nested plots is also "plot sampling"

Why not EF for expansion factor?

Since you talked about nested plots before (unequal probability) it might be better for understanding if expansion is done on the level of single trees and aggregated on plot level afterwards.

Since we take a sample of the forest area (on which we might find trees), I would rather say we upscale the plot area to a common basis. Same result but different wording...

Usually everything we measure/estimate refers to dbh height. Also the decision whether a tree is in or out is felled according to horizontal distance measured at dbh height!