In mbask/treecm: Centre of Mass Assessment and Consolidation of Trees

The centre of mass is a crucial data for arborists in order to consolidate a tree using static or dynamic cables. Given field-recorded data on branchiness of a tree, the package:

• computes and plots the centre of mass of the tree itself

• simulates the shift in CM position as branches are pruned

• computes branches slenderness coefficient in order to aid the arborist identify potentially dangerous branches

• computes the force acting on a ground plinth and its best position relating to the tree centre of mass, should the tree need to be stabilized by a steel cable

This vignette showcases some real-world cases where the package has been employed. Please notice that although the package is to be used as a quantitative aid to tree pruning and stabilization actions, the authors cannot take any responsability on the accuracy of the package results.

Usage examples

Plot centre of mass

We will make use of the data set bundled in the package to plot a basic view of masses of branches and logs of a stone pine (Pinus pinea L.) sampled in year 2010. stonePine1TreeData is a list holding a table of branch measurements, a numeric value for tree-wood density, a function for estimating branch weight, a numeric value for the position of centre of mass along branch length.

library(treecm) 
data(stonePine1TreeData)
stonePine1TreeData
class(stonePine1TreeData)

This dataset has been collected for a 17.1 metres tall stone pine whose stem was tilted approx. 20$^{\circ}$ from the vertical plane (or 80$^{\circ}$ from the horizontal plane).

The stem has been sectioned in two logs (L1 and L2), and a final branch (C). These two logs and the final branch components have been defined in the field as the "main stem" of the tree, all the other components of the tree fall into the crown. The definition of the main stem is important only for the correct assessment of the position of the anchor on the tree, should the tree need stabilization with a steel cable. Main stem components get a TRUE value in the pathToTip column.

A component with FALSE or missing value in the pathToTip column is treated as it belonging to the crown. The crown was made up of 23 branches (B1-B23), all of them horizontal (ie tilted 0$^{\circ}$).

The stonePine1TreeData dataset is a direct result of importing stonePine1FieldData with:

• wood fresh density: $\frac{kg}{m^3}$

• allometry function for branch mass: allometryABDC

• position of the centre of mass: on the branch tip

Log biomass is computed by Smalian's formula (la Marca, 2004). It is important to choose the most appropriate allometric equation in order to yield trustworthy biomass figures and, as a result, appropriate centre of mass coordinates. Allometry equations functions are discussed in section [sec:allequation], page .

The package recognizes rows that represent branches because their diameter at tip (tipD) is 0 (see more at page ).

Let's get going and compute the centre of mass of this pine:

vectors <- treeVectors(stonePine1TreeData)
CM <- centreOfMass(vectors) 
summary(CM)

The core of the package is the summary method for CM class. The centre of mass for this stone pine lies metres South-West of tree base ($^{\circ}$ from magnetic North), metres above ground. Cartesian coordinates are provided as well, though not so usefull as polar ones.

A simple visualization of tree centre of mass and its logs and branches is achieved simply by:

plot(vectors, main = "A stone pine centre of mass") 
plot(CM)

In a cartesian coordinate system whose origin lies at tree base, the masses of logs and branches are plotted as vectors pointing inwards, towards the ground. Each circle represents a branch or log mass whose radius is proportional to its mass. Likewise, the centre of mass is plotted as a vector pointing inwards, in red colour. Its height component is written alongside its label as $z$ coordinate. A red arrow approximates the direction the tree will follow should it break at its base.

It is important that, should the tree break, it does not fall onto buildings or cause damage to people. We can add buildings and other important points to the CM plot provided that we measured the polar coordinates of their relevant points, from the the tree base, using the plotPolarSegment function. Let's add a building face facing the tree:

plot(vectors, main = "A stone pine centre of mass", xlim = c(-8, 10), ylim = c(-12, 4))
plot(CM)
plotPolarSegment(210, 10.6, 140, 14.4)

Snow load

Snow may increase crown load substantially, sometimes breaking entire branches. As a side effect, snow-loaded crowns may alter tree centre of mass by moving it upwards and, in asymmetric crowns, towards the part of crown under heavier load.

Let's model a snow load that doubles the biomass of branches higher than 12 m:

rows <- substr(row.names(stonePine1TreeData$fieldData), 1, 1)
component <- substr(row.names(stonePine1TreeData$fieldData), 1, 1)
stonePine1TreeData$fieldData <- within(
  stonePine1TreeData$fieldData,
  biomass[height > 12 & component != "L"] <- biomass[height > 12 & component != "L"] * 2
  )
rm(rows)

Let's recaltulate the vectors under snow load and plot the results:

vectors <- treeVectors(stonePine1TreeData)
CM <- centreOfMass(vectors)
summary(CM)
plot(vectors, main = "2x snow load", xlim = c(-8,10), ylim =
c(-12,4))
plot(CM)
plotPolarSegment(210, 10.6, 140, 14.4)

Tree centre of mass has clearly shifted upwards but the snow load would not increase the danger on the building, should the tree collapse.

Wind load

Winds may increase load on some sectors of the crown and decrease it in other sectors. We would like to model the effect of a prevailing Southbound wind that halves branches mass in the northern sector and doubles it in the southern sector.

data(stonePine1TreeData)
rows <- substr(row.names(stonePine1TreeData$fieldData), 1, 1)
stonePine1TreeData$fieldData <- within(
  stonePine1TreeData$fieldData, {
  biomass[((azimuth >= 270 | azimuth < 90) & rows != "L")] <- biomass[((azimuth >= 270 | azimuth < 90) & rows != "L")] / 2
  biomass[((azimuth >= 90 | azimuth < 270) & rows != "L")] <- biomass[((azimuth >= 90 | azimuth < 270) & rows != "L")] * 2  
})
rm(rows)

vectors  <- treeVectors(stonePine1TreeData)
CM       <- centreOfMass(vectors)
summary(CM)

Under a heavy southbound wind the CM of the tree will move considerably towards South and 1 metre farther away from tree base. Although too simplicistic a model the results lead to the conclusion that dynamic forces in prevailing wind conditions should be taken into account when assessing tree stability.

\subsection{Effect of pruning on CM}

As far as static forces are concerned, in an effort to move centre of mass closer to tree base, we could prune a few heavy branches. Let's have a look how CM would move if we cut \texttt{B2} and \texttt{B4}:

data(stonePine1TreeData)
vectors  <- treeVectors(stonePine1TreeData)
CM       <- centreOfMass(vectors)

plot(vectors, main = "Before pruning")
plot(CM)

component <- row.names(stonePine1TreeData$fieldData)
stonePine1TreeData$fieldData$toBePruned[component %in% c("B2", "B4")] <- TRUE
vectors  <- treeVectors(stonePine1TreeData)
CM       <- centreOfMass(vectors)
plot(vectors, main = "B2 and B4 branches pruned")
plot(CM)

CM has actually moved towards tree base, and farther away from the house. As a matter of facts, branch pruning has been a slight reasonable action towards a safer tree.

Slenderness ratio {#sec:SR}

The slenderness ratio of a tree is a pure number defined as $SR = \frac{h}{d}$ where $h$ is the height of the tree trunk, and $d$ is the diameter of the tree @Matth. The SR is a measure of tree stability and is extensively used in tree stability measures carried out by Visual Tree Assessment (VTA). $SR$ in the range $30 \leq SR \leq 70$ are considered optimal, whereas $SR > 70$ lead to consider the tree at risk of breaking due to its excessive slenderness. The authors have applied the same concept to tree branches as well. While $SR$ in vertical trees has a physical meaning (Mattheck and Breloer, 1995), branches are not usually vertical. As the branch starts to deviate from the verticality (as most of the branches do) the arm of the moment gets longer, reaching a maximum limit in horizontal branches. The longer the arm, the higher the stress on the branch. In order to estimate the added stress imposed by branch angle we improved Mattheck's formula by adding a component proportional to branch tilt angle:

$SR_c = \frac{l}{d} \cdot (1 + cos \alpha)$

where $\alpha$ is branch tilt angle (i.e. $90^{\circ}$ for a vertical branch, $0^{\circ}$ for an horizontal branch). In vertical branches $SR_c=SR$, in horizontal branches $SR_c=2SR$. As far as we know this is the first attempt to apply the slenderness ratio to branches. Optimal (safe) branches could be in the range $30 \leq SR_c \leq 70$.

When stonePine1TreeData object is filled with branches length (not to be confused with tipD, the length of branch projection on the ground, from the tree base to branch tip) than $SR_c$ can be computed and plotted:

data(stonePine1TreeData) # assign length to branches
stonePine1TreeData$fieldData <- within(stonePine1TreeData$fieldData,
length[3:25] <- c(7, 7, 7, 7, 7, 7, 4, 7, 7, 4, 7, 7, 4, 7, 7, 7, 4, 7,
7, 4, 4, 7, 7)) 
vectors <- treeVectors(stonePine1TreeData) 
SR <- treeSR(stonePine1TreeData,vectors) 
plot(SR, main = "Branches slenderness ratio", xaxt='n', yaxt = 'n', xlab = '', ylab = '')

The 2D plot charts branches azimuth as arrows whose length is $SR_c$. The longer the arrows the more slender the branch. Arrows pointing inside the red circle are considered to be stable, whereas longer arrows are considered as risky ($SR_c \geq 70$). The plot may be a visual clue on the process of branch pruning selection.

Tree stabilization

Estimating the coordinates of the centre of mass of a tree is crucial to judge its static stability. The centre of mass of a perfectly balanced tree lies in between its trunk, that is the $x$ and $y$ coordinates of the CM lie inside the $\pi \cdot r^2$ surface where $r$ is the radius of tree base.

The more distant the centre of mass from tree base the higher the constrains the tree poses on the ground through its roots. When concerns about tree stability are raised and the tree needs to be consolidated a proper cabling system has to be put in place. Knowing in advance the direction the tree would fall in case of overturning is necessary to properly engineer the cabling system.

One or two static steel cables properly linking the tree to a plinth on the ground may effectively and easily lock the tree into place should its roots loose connection to the ground. Static steel cable systems include single cables and two cables systems:

• Single cable system. A single cable is layed down just opposite of CM azimuth ($\pm 180^{\circ}$). Although cheaper a solution this system has a major drawback: it does not take into consideration that trees move under wind and snow pressures, thereby shifting their CM azimuth in unpredictable ways. A single cable would not counter act tree movements and, should it break down, it would not hold the tree in place. Furthermore, as the tree grows taller, its CM shifts and the ground plinth should be moved accordingly. This is clearly not a feasible solution.

• Double cable system A system of two cables joined at the tree anchor is layed down at an even angle from the CM azimuth ($\pm 180^{\circ}$). This system holds in position a collapsed tree while allowing it some degrees of movement under snow or wind. Further, it decreases the force on the cables and on the plinth, thereby allowing for increased safety.

Since version 1.1, treecm may be used to design single cable systems.

Although putting in place the cable(s) and building an appropriate plinth on the ground are technically easy, a correct assessment of the masses into play is mandatory to design a proper system: [sec:anchor]

• The height of the anchor on the tree must be carefully chosen to be above its centre of mass (to prevent its turnover!)

• The height of the anchor on the tree must be well below its tip, to not allow the stem to flex, break and fall down

• The force acting on the cable(s) and on the plinth gets lower the farther away it is from tree base

As a rule of thumb, as far as safety is concerned, the higher on the tree the anchor is and the farther the plinth is from tree base the better. Due to the presence of other trees, buildings etc. in urban settings, scarcely ever it is possible to install very long cables, though.

Function getPlinthForce is designed to return the force on the cable and on the plinth given the length of the cable and the position of the anchor on the stem.

Its rationale lies in the comparison of the moment of the tree (applied to its centre of mass) and the moment of the anchor (applied on the anchor):

$$M_{tree}=M_{anchor}$$

Where:

$$M_{tree} = l_{cm} \cdot f_{cm} \cdot sin \alpha$$

and:

$$M_{anchor} = l_{anchor} \cdot f_{anchor} \cdot sin \beta$$

where:

• $l_x$ are the moment arms
• $f_x$ are the weight forces (masses by standard gravity)
• $\alpha$ is the angle on the centre of mass point, between the tree weight vector and the moment arm toward tree base
• $\beta$ is the angle on the anchor point, between the steel cable and the vector toward tree base

The force on the cable and on the plinth is then easily derived as:

$$f_{anchor} = \frac{M_{tree}}{l_{anchor} \cdot sin \beta}$$

Let's look at a simple example using the stonePine1TreeData dataset we seek the force and plinth position by positioning the anchor at 10m along the main stem and for a 40m long steel cable:

data(stonePine1TreeData)
vectors <- treeVectors(stonePine1TreeData) 
CM <- centreOfMass(vectors)
# We need to compute the tree moment 
treeMoment <- buildTreeMomentObject(
  centreOfMassModulus(CM),
  treeTotalBiomass(stonePine1TreeData), 
  centreOfMassAngle(CM)
  )
treeMoment <- calcMoment(treeMoment)
# We extract the logs belonging to the main stem 
mainStem <- logPathSelection(stonePine1TreeData)
(plinth <- getPlinthForce(
  l.stem = 10,
  d = 40,
  logs = mainStem,
  treeMoment = getMoment(treeMoment),
  CM = CM
  )
)

A named list of six elements is returned:

1. force ( N) is the actual force on the steel cable and plinth. Conversion to kilogram-force is approximately done by dividing it by 10 as $1 kg_F \approx 9.81 N$ (ie $kg_F$)
2. distanceOnGround (m) is the distance between the plinth and tree base (assuming a flat terrain)
3. anchorAlongStem (m) is the distance between the anchor and tree base, following the tree main stem
4. cableLength (m)
5. anchorHeight (m) is the height above ground of the anchor, equal to anchorAlongStem only when the main stem is vertical (90$^{\circ}$ above ground)
6. azimuth ($^{\circ}$) is the azimuth relative to North where the plinth should be positioned (this is simply the CM azimuth $\pm 180^\circ$)

We now have the polar coordinates for the position of the plinth (distanceOnGround, azimuth) and the force on it in order to engineer the cable width and the plinth accordingly.

What if we had constraints on the position of the plinth? It turns out that getPlinthForce is vectorized both to l.stem and d.

Let's examine the possible outcomes for a 15m to 50m long cable:

plinth <- getPlinthForce( 
  l.stem = 10, 
  d = 15:50, 
  logs = mainStem, 
  treeMoment = getMoment(treeMoment), 
  CM = CM 
  )
print(plinth$force)

There's almost a 30% decrease in the force to the plinth if we had the chance of setting the plinth 50m away from tree base. Let's now assess how force to the plinth varies when we move the anchor position along the stem. Remember that the tree centre of mass is m high, and that l.stem is the distance between tree base and the anchor, following the stem, not the anchor height on the ground:

plinth <- getPlinthForce(
    l.stem = seq(9, 12, 0.5), 
    d = 40, 
    logs = mainStem, 
    treeMoment = getMoment(treeMoment), 
    CM = CM
)
print(plinth$force)

There's almost a 20% decrease in the force to the plinth at 12m along the stem. If the tree trunk at its 12m was large enough we could position the anchor there.

Let's have a closer look at how the force to the plinth reacts by letting vary both l.stem and d:

aR <- anchorRange(mainStem, CM) 
l.stemSeq <- round(seq(aR[["z"]] + 1, aR[["hMax"]] - 2, length.out = 6), 2) 
plinth <- data.frame( 
  getPlinthForce( 
    l.stemSeq, 
    17:50, 
    mainStem,
    getMoment(treeMoment),
    CM
    )
  )
head(plinth)

We make use of the anchorRange function to select the proper range along the stem (see page ). We compute getPlinthForce for a range of distances anchor-tree base [..], each distance for a range of cable lengths [17..50]. Converting the list in a data frame let us plot it using ggplot2 package:

library(ggplot2)
ggplot(plinth, aes(x = cableLength, y = force)) +
  geom_line(aes(color = factor(anchorAlongStem), group = anchorAlongStem)) +
  scale_y_continuous('Force [N]') +
  scale_x_continuous('Cable length [m]') + 
  scale_colour_discrete("Anchor\nheight [m]") + 
  ggtitle("Force on the plinth") + 
  theme_light() 

Expressing force as a ratio to tree biomass and expressing distances relative to tree CM height or distance anchor-tree base:

plinth <- transform(
  plinth,
  distanceOverAnchorHeight = distanceOnGround / anchorAlongStem,
  heightOverAnchorHeight   = round(anchorAlongStem / CM[["z"]], 2),
  forceOverTreeBiomass     = force / treeTotalBiomass(stonePine1TreeData)/10
  )
head(plinth[c("distanceOverAnchorHeight", "heightOverAnchorHeight",
"forceOverTreeBiomass")]) 

ggplot(plinth, aes(x = distanceOverAnchorHeight, y = forceOverTreeBiomass)) + 
  geom_line() +
  facet_wrap(~heightOverAnchorHeight) +
  theme_light()

It looks like positioning the anchor $\approx 1.6$ times the height of the centre of mass, positioning the plinth 3 times the distance tree base-anchor, would expose the cable and the plith to $1/4$ of the weight of the tree itself.

[fig:collarTree]

[fig:collarCloseUp]

A word of caution

The cable system is engineered for the safety of people and their properties. Let's not forget that it should not add to the dangers of the tree breaking down! Practisioners should pay attention that:

• in the case of the tree breaking down it should immediatly lean on the anchor

• the anchor on the tree should not be strictly tightened to the trunk. The secondary growth of the trees would include the cable itself, resulting in a weakness point along the stem (figure [fig:collarDamage])

• the anchor should be loose on the trunk, positioned just above a fork. The anchor should be loosened from time to time in order to avoid the previous effect

• the cable must not be extremely tightened so as to constrain the tree in its position so as to enable it its natural movements under wind or snow. At the other end a loosened cable would enable the tree to gain speed, should it break, before leaning on the anchor. A compromise should be reached between a tight cable and a loose one

• from time to time the cable must be checked for potential damages

• as soon as the tree grows taller and changes its crown layout, or as soon as a pruning is carried out, the anchor-cable-plinth system must be modified accordingly. Rememeber that the plinth, being a quasi-permanent structure should be engineered to resist to potentially higher forces and that the anchor position should be raised as the tree get taller and heavier.

Data collection

Data collection to estimation of the centre of mass is carried out in three steps:

1. Field measurements

2. Visual check for correctness of assumptions

3. Collection of correct allometric equation in order to estimate branch and foliage biomass

Field measurements

A few field measurements are needed to estimate centre of mass position at the stem level and at the branch level. Field data are easily recorded by climbing the tree using tree-climbing techniques or by hydraulic platforms. A few instruments are needed including:

• A forestry caliper to measure diameter of logs and branches

• A clinometer, or ipsometer or any other instrument to measure height of branches or logs

• A measuring tape to measure length of branch or log projections on the ground

Measurements on logs

The stem is ideally sectioned in logs in order to compute their volume and mass. The measurements to be taken on each log include:

• Diameter at the base of the log, in cm (dBase)

• Diameter at the top of the log, in cm (dTip)

• Length of the log, in m (length)

• Azimuth of the log (azimuth), in case it is not vertical, in degrees from North (0$^{\circ}$ North, 180$^{\circ}$ South)

• Distance between tree base and the projection of log tip on the ground (tipD), in m

• Height above ground of the base of the log, in m (height)

• Log tilt (tilt) from the horizontal plane (eg a vertical log is tilted by 90$^{\circ}$, an horizontal log is tilted by 0$^{\circ}$), in degrees (optional, default to 0 if missing)

Measurements on branches

Each branch contributes to the position of the centre of mass by means of their wooden component and their foliage component. Every part of a tree carrying foliage is considered to be a branch. This definition applies to tree tip as well, although some trees may have lost their tip or have it removed during topping operations. The measurements to be taken on each branch include:

• Diameter at the base of the branch, in cm (dBase)

• Diameter at the top of the log must be 0

• Length of the branch, from its fork on the stem to its tip, in m (length), only if slenderness ratio (SR, page ) is to be computed

• Azimuth of the branch (azimuth), in degrees, usually measured with a compass (0$^{\circ}$ North, 180$^{\circ}$ South)

• Distance between tree base and the projection of branch tip on the ground (tipD), in m

• Height above ground of the branch insertion into the stem, in m (height)

• Branch tilt (tilt) from the horizontal plane (eg a vertical branch is tilted by 90$^{\circ}$, an horizontal branch is tilted by 0$^{\circ}$), in degrees (optional, default to 0 if missing)

Additional optional fields

Two more boolean fields are not strictly measured but they can initially be recorded in the field:

• Pruning status (toBePruned), which branches are going to be pruned? How would it affect CM? Optional, defaults to FALSE

• Main stem selection (pathToTip), which logs and branches make part of the "main stem" of the tree? Optional, defaults to FALSE

Visual check for correctness of assumptions

Relative position of centre of mass of branches and logs

The position of the centre of mass of a tree is computed taking into account the centre of mass of each branch and log. Pinpointing the centre of mass along a branch, taking into account branch form factor and the pattern of distribution of leaves biomass along it, would require many more field measures highering the time spent on it and the costs of the sampling.

Since the package aims to help engineering a consolidation system, the centre of mass is by default located at branches or logs tip. This leads to an estimate of the coordinates of the tree centre of mass that is farther away from the base than the actual one. This difference can be regarded as an inherent safety factor.

The package behaviour can be modified in order to let the position branches and logs centre of mass to get nearer to their base. The relative position of the centre of mass of branches and logs can be set as a real number ranging from 0.01 (base) to 1 (tip, the default behaviour). Setting can be done during import of field data using function importField and its parameter bCM or by using the setter function setBranchesCM.

Wood density

Log mass is estimated by converting its volume (as measured in the field) to fresh mass. The conversion factor is usually referred to as fresh density. Wood density is usually quite conservative among individual of the same tree species. Density values are commonly found in published literature. The dataset Dst (Niklas et al, 2010) can be a useful starting point to assess wood density, should it be unavailable, (density in $\frac{kg}{m^3}$, measured at 50% moisture content):

data(Dst)
head(Dst)

Please note that the dataset provides density figures for wood at 50% moisture content, this is not fresh density (100% moisture content) as needed by treecm. A conversion factor shall be applied by the user. As an example, a maritime pine density at 50% moisture content is:

data(Dst)
with(Dst, density[species == "Pinus pinaster"])

The datasets for stone pine provided with the package have been built taking into account a $\approx 1.07$ conversion factor from 50% to 100% moisture content density.

Choosing a correct allometric equation in order to estimate branch and foliage biomass {#sec:allequation}

It is not possible to weight the branches of a living tree. As a result branch and foliage biomass has to be estimated, this is usually done using branch diameter at its base. Models relating size or biomass to diameter of trees or branches are known as allometric equations. They usually take the form of $Y = a \cdot X^b$ where $Y$ is branch biomass, $X$ is branch diameter, $a$ and $b$ are parameters estimated on a sample of branches (eg during a pruning process).

When sampling is not possible one should rely on published allometric equations and feed them to treecm. Currently treecm ships with four allometric equations:

• allometryABDC, tested on stone pine branches, 5-16 cm diameter, returns fresh weight

• allometryAsca2011, tested on stone pine branches, 8-16 cm diameter, predominantly from the lower layers of the crown, returns fresh weight, it is based on a subsample of allometryABDC and it must be considered as deprecated

• allometryCutini2009, tested on stone pine trees (not on branches), 24+ cm diameter, returns biomass, dry weight, (Cutini et al, 2009), its results should be increased by the estimated amount of water present in the branches

• allometryPorte2002, tested on maritime pine branches, 10- cm diameter, returns biomass (dry weight), (Porté et al, 2002), its results should be increased by the estimated amount of water present in the branches

The proper allometric equation to be used must be fed to treecm when importing field data using function importFieldData, parameter branchesAllometryFUN. Please notice that:

• one should pick an allometric equation that yields fresh mass of branches in order to get results as closer as possible to the real tree centre of mass

• choosing the allometric equation is crucial to a correct estimation of masses. An allometric equation must be choosen according mainly to the tree species, and to the range of branch diameters it has been fitted on

A real example

Let's have a look at how the allometric equation may affect the estimate of branch masses in a stone pine. Dataset stonePine2FieldData holds field recorded value for a $\approx 11$m tall stone pine, having only a few very big branches. We will compare the estimate of whole tree mass carried out applying two allometric equations: allometryCutini2009 ($Y=-198 + 0.620 \cdot X^2$) and allometryABDC ($Y=0.17 \cdot X^{2.4}$).

csvFile <- system.file("doc", "CopyOfstonePine2FieldData.csv", package = "treecm")
sP.Cutini <- treeBiomass(importFieldData(csvFile, 650, allometryCutini2009))
head(sP.Cutini$fieldData)
sP.ABDC   <- treeBiomass(importFieldData(csvFile, 650, allometryABDC))
head(sP.ABDC$fieldData)

biomassRaw <- data.frame(
  cutini = sP.Cutini$fieldData$biomass,
  ABDC   = sP.ABDC$fieldData$biomass,
  code   = rownames(sP.ABDC$fieldData), 
  diameter = sP.ABDC$fieldData$dBase
 )

biomass <- reshape2::melt(
 biomassRaw
 , measure.vars = c("ABDC", "cutini")
 , value.name = "biomass"
 , variable.name = "allometry"
)

rm(biomassRaw)

(treeBiomass <- with(biomass, tapply(biomass, allometry, sum)))

allometryABDC function estimate for the biomass of the tree is r round(treeBiomass[["ABDC"]]/treeBiomass[["cutini"]], 1) times higher than allometryCutini2009 function! This huge gap is accounted for by the way the allometric equations were designed:

1. allometryABDC has been fitted on small-diameter branches, as a result it may predict unreliable mass figures for out of range branches
2. allometryCutini2009 has been fitted on 24+ cm trees, as a result it properly describes the relationship between mass and diameter as far as diameter is concerned. We must assume that the relationship tree diameter-tree mass holds for branches as well, though. Further the equation yields biomass at 0% moisture content, a figure much lower than fresh biomass.

The difference in the estimates may be clearly appreciated in the following figure.

ggplot(biomass, aes(x = code, y = biomass)) + 
  geom_bar(aes(fill = allometry), position = "dodge", stat = "identity") +
  theme_light()

ggplot(biomass, aes(x = diameter, y = biomass)) + 
  geom_point(aes(shape = allometry), size = 5) +
  theme_light()

Notice that cutini's biomass estimate for B4 is unreasonably negative. This is due to the small diameter branch (10 cm), far below the 24cm lower limit. Also notice consitently lower estimates from cutini's function.

Which function to use? The answer is constrained by the fact that ABDC function cannot be reasonably applied since it has not been fitted on the range of branch diameters our pine shows. How shall we adapt cutini's estimates? First of all we need to fix the biomass estimate for B4, we shall use the ABDC biomass figure for it. Then we shall convert 0% moisture biomass to fresh biomass. We will use a $1.5$ expansion factor assuming there's a 50% decrease in mass from fresh to dry state.

biomass <- within(biomass, {
  biomass[allometry == "cutini" & code != "L1"] <- biomass[allometry == "cutini" & code != "L1"] * 1.5
  biomass[allometry == "cutini" & code == "B4"] <- biomass[allometry == "ABDC" & code == "B4"]
})

with(biomass, tapply(biomass, allometry, sum))

ggplot(biomass, aes(x = diameter, y = biomass)) + 
  geom_point(aes(shape = allometry), size = 5) +
  theme_light()

This is just an attempt to raise awarness on the complexities of choosing a correct allometric equation. When we reckon our data are correct we just have to plug them into a proper fieldData data frame:

sP.Cutini$fieldData$biomass <- biomass$biomass[biomass$allometry == "cutini"]
head(sP.Cutini$fieldData)

Correct layout of CSV file

A sample CSV data file is provided in the data directory. Function importFieldData loads and stores CSV files and along with needed data. CSV files are made up of 10 columns. The first row has to hold column headers. Headers are case sensitive. Each row holds individual log or branch data. Headers include:

1. code a simple code assigned to each log or branch
2. azimuth orientation, ie: compass bearing in degrees
3. dBase diameter of log or branch basal section, in cm
4. dTip diameter of log or branch tip (always 0 for branches), in cm
5. length log length, in m; also branch length if slenderness ratio (SR, page ) is to be computed, leave it empty otherwise
6. tipD distance of the tip of the log or branch to tree base (different from branch length when tree stem is not vertical)
7. height height of log basal section of height of branch insertion on stem
8. tilt log or branch tilt from the horizontal plane (eg a vertical branch is tilted by 90$^{\circ}$, an horizontal branch is tilted by 0$^{\circ}$), in degrees (optional, only useful to estimate $z$ coordinate of centre of mass)
9. toBePruned boolean to simulate branch pruning
10. pathToTip boolean to point out the "main stem"

Rules to layout a correct CSV file {#sec:rules}

Please notice that some rules have to be followed in order to record sound data in the field:

• the diameter of the tip of L1 is equal to the diameter of the base of L2. L2 tip diameter is, in turn, equal to C base diameter. Height figures must match between consecutive logs, as well as diameter measures do
• the distance of the tip of the branch (tipD) is not the length of the branch but the distance between tree base (the origin of the cartesian plot) and the branch tip
• note that length has been only recorded for the C branch (not considering logs) as it is the only branch not being horizontal. Non horizontal branches affect tree CM z-coordinate. When non-horizontal branches are present, and if one is interested in the z-coordinate of CM, than branch length and its angle from the horizonatl plane (tilt) should also be recorded. Otherwise branch length is not needed.

Contribute!

treecm is an ongoing project hosted on GitHub. Many areas need to be expanded including:

• branch biomass estimation; allometric equations are used to estimate fresh branch and foliage biomass. So far only branch biomass for stone pine and maritime pine have been developed or integrated into the software from published data. We need to expand further the number of species represented (allometric equations relating branches fresh weight and their diameter, or raw data), particularly for the common species in urban areas such as cedars, magnolias, oaks
• The package does not estimate the position of the centre of mass of tree branches. This position may vary according to foliage mass and its distribution along the branchs, branch tapering, quantity of water in leaves (ie shaded or lit leaves) etc. The position must be fed to the package during data loading, as the variable bCM. Although going for the safe road, setting a branch centre of mass position on its tip may not be sufficiently precise should one assess wood quality as a function of load balance. Work is under way in order to to model branch load balance
• As far as postion of centre of mass, the package does not tell branches and logs apart. The position of CM in logs follows branches CM position settings, though not realistic

Future enhancements

• getPlinthForce should also account for the case where two cables could be laid down from the same anchor, to enhance safety
• getPlinthForce could also be used to test the engineering of a laid down system

Bibliography

Cutini, A. and Hajny, M. and Gugliotta, O. and Manetti, M. and Amorini, E. 2009, Effetti della struttura del popolamento sui modelli di stima del volume e della biomassa epigea (Pineta di Castelfusano - Roma) Forest@, 6, 75–84

la Marca, O. 2004, Elementi di dendrometria. Patron Editore (Bologna), p. 119

Niklas, K. J. and Spatz, H.-C., 2010, Worldwide correlations of mechanical properties and green wood density American Journal of Botany, 97, 1587–1594

Porté, A. and Trichet, P. and Bert, D. and Loustau, D. 2002, Allometric relationships for branch and tree woody biomass of Maritime pine (Pinus pinaster Ait.) Forest Ecology and Management, 158, 71–83

Mattheck, C. and Breloer, H., 1995, The Body Language of Trees: A Handbook for Failure Analysis (Research for Amenity Trees). HMSO (London).

mbask/treecm documentation built on May 21, 2019, 2:26 p.m.