Introduction to crownmetrics

Notation

Throughout this document the following symbols are used:

1.1 Ellipsoid

The ellipsoid is the most widely used approximation for broadleaf and urban trees. When \(CW = CL\) the formula reduces to a sphere.

\[V_{\text{ellipsoid}} = \frac{4}{3}\,\pi \left(\frac{CL}{2}\right)\left(\frac{CW}{2}\right)^{2}\]

crown_volume_ellipsoid(crown_width = 3.5, crown_length = 5.0)
#> [1] 32.07043

1.2 Cone

Suitable for conifers with narrow, pointed crowns (e.g. Pinus spp., Picea spp.).

\[V_{\text{cone}} = \frac{1}{3}\,\pi \left(\frac{CW}{2}\right)^{2} CL\]

crown_volume_cone(crown_width = 3.5, crown_length = 5.0)
#> [1] 16.03521

1.3 Cylinder

Represents trees with a uniform crown width from base to top.

\[V_{\text{cylinder}} = \pi \left(\frac{CW}{2}\right)^{2} CL\]

crown_volume_cylinder(crown_width = 3.5, crown_length = 5.0)
#> [1] 48.10564

1.4 Paraboloid

An intermediate shape between the cone and the cylinder, often considered a reasonable approximation for broadleaf trees.

\[V_{\text{paraboloid}} = \frac{1}{2}\,\pi \left(\frac{CW}{2}\right)^{2} CL\]

crown_volume_paraboloid(crown_width = 3.5, crown_length = 5.0)
#> [1] 24.05282

1.5 Fan / Umbrella

Used for trees with wide, flat crowns (e.g. Pinus pinea). The vertical extent of the crown solid is approximated by the stem diameter at breast height converted to meters (\(DBH / 100\)).

\[V_{\text{fan}} = \frac{\pi\,CL^{2}}{4} \cdot \frac{DBH}{100}\]

crown_volume_fan(crown_length = 5.0, dbh = 22.0)
#> [1] 4.31969

1.6 Comparing shapes

The relationship between shapes is fixed for given \(CW\) and \(CL\): \(V_{\text{cylinder}} = 2\,V_{\text{paraboloid}} = 3\,V_{\text{cone}}\), with the ellipsoid falling between the paraboloid and the cylinder.

cw <- 3.5   # crown width (m)
cl <- 5.0   # crown length (m)

volumes <- c(
  ellipsoid  = crown_volume_ellipsoid(cw, cl),
  cone       = crown_volume_cone(cw, cl),
  cylinder   = crown_volume_cylinder(cw, cl),
  paraboloid = crown_volume_paraboloid(cw, cl)
)

round(volumes, 2)
#>  ellipsoid       cone   cylinder paraboloid 
#>      32.07      16.04      48.11      24.05

1.7 Unified dispatcher

crown_volume() selects the model via its shape argument, accepting "ellipsoid", "cone", "cylinder", "paraboloid", or "fan":

crown_volume(crown_width = 3.5, crown_length = 5.0, shape = "ellipsoid")
#> [1] 32.07043
crown_volume(crown_width = 3.5, crown_length = 5.0, shape = "cone")
#> [1] 16.03521
crown_volume(crown_length = 5.0, dbh = 22.0, shape = "fan")
#> [1] 4.31969

2.1 Crown profile areas (lateral cross-section)

These represent the two-dimensional silhouette of the crown as seen from the side (McPherson & Rowntree, 1988; Zhu, Kleinn & Nölke, 2021).

Ellipse — used with the ellipsoid and paraboloid volume models:

\[A_{\text{ellipse}} = \pi \left(\frac{CL}{2}\right)\left(\frac{CW}{2}\right)\]

Triangle — used with the cone volume model:

\[A_{\text{triangle}} = \frac{CW \cdot CL}{2}\]

Rectangle — used with the cylinder volume model:

\[A_{\text{rectangle}} = CW \cdot CL\]

Fan / umbrella — used with the fan volume model:

\[A_{\text{fan}} = \frac{\pi\,CL}{4}\]

crown_area(crown_width = 3.5, crown_length = 5.0, shape = "ellipse")
#> [1] 13.74447
crown_area(crown_width = 3.5, crown_length = 5.0, shape = "triangle")
#> [1] 8.75
crown_area(crown_width = 3.5, crown_length = 5.0, shape = "rectangle")
#> [1] 17.5
crown_area(crown_length = 5.0, shape = "fan")
#> [1] 3.926991

2.2 Crown projection area (horizontal)

The crown projection area (CPA) is the area of the crown’s shadow on the ground, assuming a circular crown outline (Sayn-Wittgenstein & Aldred, 1972):

\[CPA = \pi \left(\frac{CW}{2}\right)^{2}\]

crown_projection_area(crown_width = 3.5)
#> [1] 9.621128

3.1 Crown Ratio — CR

The fraction of total height occupied by the living crown. Higher values indicate greater photosynthetic potential and crown vitality.

\[CR = \frac{CL}{H}\]

crown_ratio(crown_length = 5.0, total_height = 18.0)
#> [1] 0.2777778
crown_ratio(crown_length = 5.0, total_height = 18.0, as_percentage = TRUE)
#> [1] 27.77778

3.2 Crown Form Factor — CF

The ratio of crown width to crown length. Values \(> 1\) indicate wide, flat crowns; values \(< 1\) indicate tall, narrow crowns.

\[CF = \frac{CW}{CL}\]

crown_form(crown_width = 3.5, crown_length = 5.0)
#> [1] 0.7

3.3 Slenderness Index — SLI

The ratio of total height to DBH (both expressed in meters). Higher values indicate more slender trees with greater susceptibility to wind and snow damage.

\[SLI = \frac{H}{DBH / 100}\]

slenderness(total_height = 18.0, dbh = 22.0)
#> [1] 81.81818

3.4 Salience Index — SAI

Expresses how many times wider the crown is than the trunk diameter (both in meters). Reflects the tree’s capacity to occupy horizontal space relative to its stem size.

\[SAI = \frac{CW}{DBH / 100}\]

salience_index(crown_width = 3.5, dbh = 22.0)
#> [1] 15.90909

3.5 Scope Index — SCI

The ratio of crown width to total tree height. Describes the lateral competitive reach of the crown relative to tree height.

\[SCI = \frac{CW}{H}\]

scope_index(crown_width = 3.5, total_height = 18.0)
#> [1] 0.1944444

3.6 Vital Space Index — VSI

The ratio of crown projection area to stem basal area. Expresses how many times more ground area the crown occupies compared to the stem cross-section. Note that the \(\pi\) terms cancel, so \(VSI = SAI^{2}\).

\[VSI = \frac{CPA}{BA} = \frac{\pi\,(CW/2)^{2}}{\pi\,(DBH/200)^{2}} = \left(\frac{CW}{DBH/100}\right)^{2}\]

vital_space_index(crown_width = 3.5, dbh = 22.0)
#> [1] 253.0992

3.7 Summary table

3.8 Computing all indices at once

crown_morphometrics(
  crown_width  = 3.5,
  crown_length = 5.0,
  total_height = 18.0,
  dbh          = 22.0
)
#>   crown_ratio crown_form slenderness salience_index scope_index
#> 1   0.2777778        0.7    81.81818       15.90909   0.1944444
#>   vital_space_index
#> 1          253.0992