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Software for spatially-explicit simulation of forest dynamics

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The non-spatial density dependent crown depth and radius relationships

The density dependent equations for crown radius and crown depth use non-spatial measures of density to influence crown radius and crown depth. Density is measured across the plot as a whole, not locally (thus "non-spatial").

Crown Radius Parameters

Parameter name	Description
Non-Spatial Density Dep. Inst. Crown Height "a"	The "a" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "b"	The "b" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "c"	The "c" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "d"	The "d" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "e"	The "e" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "f"	The "f" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "g"	The "g" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "h"	The "h" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "i"	The "i" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Density Dep. Inst. Crown Height "j"	The "j" term in the instrumental crown depth equation, used to calculate crown radius
Non-Spatial Exp. Density Dep. Crown Radius "D1"	The "D1" term
Non-Spatial Exp. Density Dep. Crown Radius "a"	The "a" term
Non-Spatial Exp. Density Dep. Crown Radius "b"	The "b" term
Non-Spatial Exp. Density Dep. Crown Radius "c"	The "c" term
Non-Spatial Exp. Density Dep. Crown Radius "d"	The "d" term
Non-Spatial Exp. Density Dep. Crown Radius "e"	The "e" term
Non-Spatial Exp. Density Dep. Crown Radius "f"	The "f" term

Crown Depth Parameters

Parameter name	Description
Non-Spatial Density Dep. Inst. Crown Radius "a"	The "a" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "b"	The "b" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "c"	The "c" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "d"	The "d" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "e"	The "e" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "f"	The "f" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "g"	The "g" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "h"	The "h" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "i"	The "i" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Density Dep. Inst. Crown Radius "j"	The "j" term in the instrumental crown radius equation, used to calculate crown depth
Non-Spatial Log. Density Dep. Crown Height "a"	The "a" term
Non-Spatial Log. Density Dep. Crown Height "b"	The "b" term
Non-Spatial Log. Density Dep. Crown Height "c"	The "c" term
Non-Spatial Log. Density Dep. Crown Height "d"	The "d" term
Non-Spatial Log. Density Dep. Crown Height "e"	The "e" term
Non-Spatial Log. Density Dep. Crown Height "f"	The "f" term
Non-Spatial Log. Density Dep. Crown Height "g"	The "g" term

In addition to the use of density variables, the density dependent equations for crown width uses an estimate of crown depth as a dependent variable (and vice versa). This estimated value of crown width and crown depth (rad_i and ch_i) used in the density dependent equations come from the instrumental variable equations. Calculating the instrumental variables equations avoids "uncoupling" the crown radius - crown depth relationship.

The non-spatial exponential density dependent crown radius function is:

rad = D1 * DBH ^a * Height ^b * ch_i ^c * STPH ^d * BAPH ^e * BAL ^f

where:

rad is the crown radius, in meters
D1 is the Non-Spatial Exp. Density Dep. Crown Radius "D1" parameter
a is the Non-Spatial Exp. Density Dep. Crown Radius "a" parameter
b is the Non-Spatial Exp. Density Dep. Crown Radius "b" parameter
c is the Non-Spatial Exp. Density Dep. Crown Radius "c" parameter
d is the Non-Spatial Exp. Density Dep. Crown Radius "d" parameter
e is the Non-Spatial Exp. Density Dep. Crown Radius "e" parameter
f is the Non-Spatial Exp. Density Dep. Crown Radius "f" parameter
DBH is the tree's DBH, in cm
Height is the tree height, in meters
ch_i is the instrumental crown depth of the target tree, in meters, calculated using the function below
STPH is number of stems per hectare of adult trees within the entire plot
BAPH is the basal area, in m² per hectare, of adult trees within the entire plot
BAL is the sum of the basal area of all trees taller than the height of the target tree, in m² per hectare

The instrumental equation for calculating ch_i is as follows:

ch_i = a + b * DBH + c * Height + d * DBH ² + e * Height ² + f / DBH + g * STPH + h * BAPH + i * BAL + j * (Height / DBH)

where:

a is the Non-Spatial Density Dep. Inst. Crown Height "a" parameter
b is the Non-Spatial Density Dep. Inst. Crown Height "b" parameter
c is the Non-Spatial Density Dep. Inst. Crown Height "c" parameter
d is the Non-Spatial Density Dep. Inst. Crown Height "d" parameter
e is the Non-Spatial Density Dep. Inst. Crown Height "e" parameter
f is the Non-Spatial Density Dep. Inst. Crown Height "f" parameter
g is the Non-Spatial Density Dep. Inst. Crown Height "g" parameter
h is the Non-Spatial Density Dep. Inst. Crown Height "h" parameter
i is the Non-Spatial Density Dep. Inst. Crown Height "i" parameter
j is the Non-Spatial Density Dep. Inst. Crown Height "j" parameter
DBH is the tree's DBH, in cm
Height is the tree height, in meters
STPH is number of stems per hectare of adult trees within the entire plot
BAPH is the basal area, in m² per hectare, of adult trees within the entire plot
BAL is the sum of the basal area of all trees taller than the height of the target tree, in m² per hectare

The non-spatial logistic density dependent crown depth function is:

where:

ch is the crown depth, in meters
height is the tree's height, in m
a is the Non-Spatial Log. Density Dep. Crown Height "a" parameter
b is the Non-Spatial Log. Density Dep. Crown Height "b" parameter
c is the Non-Spatial Log. Density Dep. Crown Height "c" parameter
d is the Non-Spatial Log. Density Dep. Crown Height "d" parameter
e is the Non-Spatial Log. Density Dep. Crown Height "e" parameter
f is the Non-Spatial Log. Density Dep. Crown Height "f" parameter
g is the Non-Spatial Log. Density Dep. Crown Height "g" parameter
DBH is the tree's DBH, in cm
rad_i is the instrumental crown radius of the target tree, in meters, calculated using the function below
STPH is number of stems per hectare of adult trees within the entire plot
BAPH is the basal area, in m² per hectare, of adult trees within the entire plot
BAL is the sum of the basal area of all trees taller than the height of the target tree, in m² per hectare

The instrumental equation for calculating rad_i is as follows:

rad_i = a + b * DBH + c * Height + d * DBH ² + e * Height ² + f / DBH + g * STPH + h * BAPH + i * BAL + j * (Height / DBH)

where:

a is the Non-Spatial Density Dep. Inst. Crown Radius "a" parameter
b is the Non-Spatial Density Dep. Inst. Crown Radius "b" parameter
c is the Non-Spatial Density Dep. Inst. Crown Radius "c" parameter
d is the Non-Spatial Density Dep. Inst. Crown Radius "d" parameter
e is the Non-Spatial Density Dep. Inst. Crown Radius "e" parameter
f is the Non-Spatial Density Dep. Inst. Crown Radius "f" parameter
g is the Non-Spatial Density Dep. Inst. Crown Radius "g" parameter
h is the Non-Spatial Density Dep. Inst. Crown Radius "h" parameter
i is the Non-Spatial Density Dep. Inst. Crown Radius "i" parameter
j is the Non-Spatial Density Dep. Inst. Crown Radius "j" parameter
DBH is the DBH of the tree, in cm
Height is the tree height, in meters
STPH is number of stems per hectare of adult trees within the entire plot
BAPH is the basal area, in m² per hectare, of adult trees within the entire plot
BAL is the sum of the basal area of all trees taller than the height of the target tree, in m² per hectare