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\title{An Introduction to treePlotArea}
\author{Andreas Dominik Cullmann}

\SweaveOpts{echo=false}
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\begin{document}
\maketitle
\tableofcontents

\section{What treePlotArea Does}
The German national forest inventory uses angle count
sampling, a sampling method invented by Bitterlich in 1947
(\cite{bitterlich1947}) and
extended by Grosenbaugh \cite{grosenbaugh1952} as probability proportional to
size sampling.
When plots are located near stand boundaries, their sizes and hence
their probabilities need to be corrected.

\subsection{Why corrections?}
As you can see in Figure \ref{fig:bwi2022de}, a screenshot from \texttt{bwi2022de},
the software used for the field surveys of the German national forest inventory in 2022,
the tree plot area of tree 7 of corner 2 of tract 166 gets intersected by two
stand boundaries.
So its plot area needs to be corrected.

\begin{figure}[h!]
  \centering
  \includegraphics{teb_166_2_7.png}
  \caption{Tree plot from \texttt{bwi2022de}.}
  \label{fig:bwi2022de}
\end{figure}
\subsection{What did we do before?}
Until now, the German national forest inventory used a program called
\texttt{grenzkreis} that was written by Bernhard B\"o{}sch at the Forest Research
Institute of Baden-W\"u{}rttemberg.
It is written in C++, poorly documented and, to our limited understanding, hard to maintain.
The last version dates from March 2011.
We felt the need to re-write the library in R.
Gerald K\"a{}ndler used \texttt{grenzkreis} to provide reference data to this package, which we use for testing.

\subsection{Single Tree Plot Areas}
If we give a tract, corner and tree id, we can plot the effective tree plot area
and output the ratio of the theoretical (red) to the effective (green) tree plot area:

<<ptpa, echo = TRUE, eval = TRUE, fig=FALSE>>==
library("treePlotArea")
tnr <- 166
enr <- 2
bnr <- 7
angle_counts <- bw2bwi2022de(get(data("trees",
                                      package = "treePlotArea")))
angle_counts <- select_valid_angle_count_trees(angle_counts)
boundaries <- get(data("boundaries", package = "treePlotArea"))
cf <- plot_tree_plot_area(angle_counts = angle_counts,
                          boundaries = boundaries,
                          tnr =  tnr, enr = enr, bnr = bnr,
                          frame_factor = 0.35)
@

<<echo = FALSE, eval = TRUE, fig = TRUE>>==
<<ptpa>>
@

Now we compare the correction factor to the reference value given by
\texttt{grenzkreis}:
<<cf, echo = TRUE, eval = TRUE, fig=FALSE>>==
print(cf)
angle_counts[angle_counts[["tnr"]] == tnr &
             angle_counts[["enr"]] == enr &
             angle_counts[["bnr"]] == bnr, "kf2"]
@
This is rather close, the difference probably due to different approximations
of circles by polygons in the two libraries/packages.


\subsection{Tree Plot Areas for Forest Inventories}
Of course we do not look at single trees, we want to get the correction factors
for collections of trees.

<<nrow, echo = TRUE, eval = TRUE, fig=FALSE>>==
nrow(angle_counts)
nrow(boundaries)
@

<<cf, echo = TRUE, eval = TRUE, fig=FALSE>>==
correction_factors <- get_correction_factors(angle_counts,
                                             boundaries,
                                             verbose = FALSE)
@
%  <<cft, echo = FALSE, eval = TRUE, fig=FALSE, results=verbatim, print = FALSE>>==
%  # Fake the above warning swallowed up by Sweave
%  w <- tryCatch(
%                correction_factors <- get_correction_factors(angle_counts,
%                                                             boundaries,
%                                                             verbose = FALSE),
%                warning = identity
%  )
%  cat("Warning message:", w$message, sep = "\n")
%  @
%  There's a warning above about an invalid boundary.
%  It is the one in
%  tract 2607, corner 2: azimuth values in \texttt{spa\_gon} and
%  \texttt{spk\_gon} are identical:
%
%  <<bo, echo = TRUE, eval = TRUE, fig=FALSE>>==
%  subset(boundaries, tnr == 2607)
%  @
%  That means it runs exactly through the plot.
%  Anyway, since
%  \texttt{grenzkreis} dealt with it, we do that too (although we shouldn't, see \\
%  \texttt{help("get\_boundary\_polygons", package = "treePlotArea")} \\for
%  more on
%  this):
%  <<cf, echo = TRUE, eval = TRUE, fig=FALSE>>==
%  print(subset(correction_factors, tnr == 2607 & enr == 2))
%  @


\section{Customizing Data}
We have used data coming with package: one data frame containing angle counts
(trees), and one containing boundaries.

The names of the data columns essential to this package are retrieved from the
global
option \texttt{treePlotArea}.
We can set it to its default (which is done by the package wherever needed
without overwriting an slots in that list already set) and then inspect it:

<<options, echo = TRUE, eval = TRUE>>==
set_options()
str(getOption("treePlotArea"))
@


The columns of our data frames could have different names (we just convert them
to upper case here):
<<data_rename, echo = TRUE, eval = TRUE>>==
names(angle_counts) <- toupper(names(angle_counts))
names(boundaries) <- toupper(names(boundaries))
@

We would then have to set the options accordingly
<<data_options, echo = TRUE, eval = TRUE>>==
option_list <- sapply(get_defaults(), function(x) lapply(x, toupper))
set_options(angle_counts = option_list[["angle_counts"]],
            boundaries = option_list[["boundaries"]])
@

and the results stay the same:
<<data_runit, echo = TRUE, eval = TRUE>>==
correction_factors_upper <- get_correction_factors(angle_counts,
                                                   boundaries,
                                                   verbose = FALSE)
RUnit::checkEquals(correction_factors_upper, correction_factors,
                   checkNames = FALSE)
@


\section{How treePlotArea Works}
This is the rather technical part, you might not be interested \ldots

\subsection{Flexed Boundaries}
First, we decide whether a flexed boundary is flexed towards the center of the
corner or away from it:

<<ptpa_1, echo = TRUE, eval = TRUE, fig=TRUE>>==
cf <- plot_tree_plot_area(angle_counts = angle_counts,
                          boundaries = boundaries,
                          tnr =  tnr, enr = enr, bnr = bnr,
                          frame_factor = 1)
@

Yellow lines mark boundaries flexed away from the center,
blue ones boundaries flexed towards it.

\paragraph{How do we decide whether a flexed boundary is `flexed towards the center`?}
We build triangles from each flexed boundary that intersect a circle with a
radius of 50 km. If the tree coordinates do  fall into that triangle, the
boundary is  `flexed towards the center`.


\subsection{Boundary Polygons}
We then build polygons (depending on the type of boundary):
<<ptpa_4, echo = TRUE, eval = TRUE, fig=TRUE>>==
cf <- plot_tree_plot_area(angle_counts = angle_counts,
                          boundaries = boundaries,
                          tnr =  tnr, enr = enr, bnr = bnr,
                          frame_factor = 4)
@

Boundaries flexed towards the center are split into two straight lines.
For each straight line, a square `away from the center` is defined.
For boundaries flexed away, pentagon `away from the center` are defined.

`Away from the center` means that the polygons are created such that their
corner towards the center lie on a circle with radius 100 meter.
We choose this value because for a tree with a breast height diameter of 2000
millimeter, the tree plot area for angle count factor 4 (which is used in the
German national forest inventory) will be a circle with a radius of 50m. We just
double that value to make sure no tree plot area could possibly stick out of our
polygons somewhere.
We then use package \texttt{sf} to intersect the tree plot area with the
boundary polygons.
That's it.

\bibliography{bibliography}
\end{document}
