Allometry
In this document:
Allometry parameters
The "standard" crown depth and radius relationships
The "Chapman-Richards" crown depth and radius relationships
The non-spatial density dependent crown depth and radius relationships
The NCI crown depth and radius relationships
DBH - diameter at 10 cm relationship
The "standard" diameter-height relationships
The "linear" diameter-height relationship
The "reverse linear" diameter-height relationship
The "power" diameter-height relationship
Allometry is the relationships between various aspects of a tree's size and shape. A tree may use different relationships for different life history stages.
You can choose the relationship used by each life history stage of each species. These can be freely mixed-and-matched. Use the Edit Allometry Functions window to set the allometry functions.
Definitions: DBH (diameter at breast height) is the diameter of a tree trunk at 1.35 meters above the ground. Diameter at 10 cm, or diam10, is the diameter of a tree trunk 10 cm above the ground.
In general, crowns are modeled as cylinders, with a radius and a height. Specific behaviors may make different assumptions but if so they should be clearly stated in that behavior's documentation.
Seedlings in SORTIE-ND do not have crowns. Saplings and adults (and, in some cases, snags) all use the same relationships to describe crown shape.
Allometry parameters
- Adult Linear Function Intercept The intercept of the adult linear function for DBH and height.
- Adult Linear Function Slope The slope of the adult linear function for DBH and height.
- Adult Reverse Linear Function Intercept The intercept of the adult reverse linear function for DBH and height.
- Adult Reverse Linear Function Slope The slope of the adult reverse linear function for DBH and height.
- Chapman-Richards Asymptotic Crown Height The asymptotic crown depth (or length), in m, of the Chapman-Richards crown depth equation.
- Chapman-Richards Asymptotic Crown Radius The asymptotic crown radius, in m, of the Chapman-Richards crown radius equation.
- Chapman-Richards Crown Height Intercept The intercept of the Chapman-Richards crown depth equation. This represents the crown depth, in m, of the smallest possible sapling.
- Chapman-Richards Crown Height Shape 1 (b) The first shape parameter, b, of the Chapman-Richards crown depth equation.
- Chapman-Richards Crown Height Shape 2 (c) The second shape parameter, c, of the Chapman-Richards crown depth equation.
- Chapman-Richards Crown Radius Intercept The intercept of the Chapman-Richards crown radius equation. This represents the crown radius, in m, of the smallest possible sapling.
- Chapman-Richards Crown Radius Shape 1 (b) The first shape parameter, b, of the Chapman-Richards crown radius equation.
- Chapman-Richards Crown Radius Shape 2 (c) The second shape parameter, c, of the Chapman-Richards crown radius equation.
- Crown Height Exponent The exponent in the standard equation for calculating crown depth.
- Crown Radius Exponent The exponent in the standard equation for determining the crown radius.
- Intercept of DBH to Diameter at 10 cm Relationship The intercept of the linear relationsip between the DBH, in cm, and the diameter at 10 cm height, in cm, in small trees. Used by all species.
- Maximum Tree Height, in meters The maximum tree height for a species, in meters. No tree, no matter what allometric function it uses, is allowed to get taller than this. Used by all species.
- NCI Crown Depth - Alpha NCI function exponent. Used to calculate NCI crown depth.
- NCI Crown Depth - Beta NCI function exponent. Used to calculate NCI crown depth.
- NCI Crown Depth - Crowding Effect "n" Crowding effect exponent. Used to calculate NCI crown depth.
- NCI Crown Depth - Gamma NCI function exponent. Used to calculate NCI crown depth.
- NCI Crown Depth Lambda for Species X Neighbors The competitive effect of neighbors of species X. Used to calculate NCI crown depth.
- NCI Crown Depth - Max Potential Depth (m) The maximum possible value for crown depth, in m. Used to calculate NCI crown depth.
- NCI Crown Depth - Max Search Distance for Neighbors (m) The maximum distance, in m, at which a neighboring tree has competitive effects on a target tree. Used to calculate NCI crown depth.
- NCI Crown Depth - Minimum Neighbor DBH (cm) The minimum DBH for trees of that species to compete as neighbors. Values are needed for all species. Used to calculate NCI crown depth.
- NCI Crown Depth - Size Effect "d" Size effect function exponent. Used to calculate NCI crown depth.
- NCI Crown Radius - Alpha NCI function exponent. Used to calculate NCI crown radius.
- NCI Crown Radius - Beta NCI function exponent. Used to calculate NCI crown radius.
- NCI Crown Radius - Crowding Effect "n" Crowding effect exponent. Used to calculate NCI crown radius.
- NCI Crown Radius - Gamma NCI function exponent. Used to calculate NCI crown radius.
- NCI Crown Radius Lambda for Species X Neighbors The competitive effect of neighbors of species X. Used to calculate NCI crown radius.
- NCI Crown Radius - Max Potential Radius (m) The maximum possible value for crown radius, in m. Used to calculate NCI crown radius.
- NCI Crown Radius - Max Search Distance for Neighbors (m) The maximum distance, in m, at which a neighboring tree has competitive effects on a target tree. Used to calculate NCI crown radius.
- NCI Crown Radius - Minimum Neighbor DBH (cm) The minimum DBH for trees of that species to compete as neighbors. Values are needed for all species. Used to calculate NCI crown radius.
- NCI Crown Radius - Size Effect "d" Size effect function exponent. Used to calculate NCI crown radius.
- Non-Spatial Density Dep. Inst. Crown Height "a" The "a" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "b" The "b" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "c" The "c" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "d" The "d" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "e" The "e" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "f" The "f" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "g" The "g" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "h" The "h" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "i" The "i" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Density Dep. Inst. Crown Height "j" The "j" term in the instrumental crown depth equation, used to calculate crown radius for the non-spatial exponential density dependent function.
- Non-Spatial Exp. Density Dep. Crown Radius "D1" The "D1" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "a" The "a" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "b" The "b" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "c" The "c" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "d" The "d" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "e" The "e" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Exp. Density Dep. Crown Radius "f" The "f" term in the non-spatial exponential density dependent crown radius function.
- Non-Spatial Density Dep. Inst. Crown Radius "a" The "a" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "b" The "b" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "c" The "c" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "d" The "d" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "e" The "e" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "f" The "f" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "g" The "g" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "h" The "h" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "i" The "i" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Density Dep. Inst. Crown Radius "j" The "j" term in the instrumental crown radius equation, used to calculate crown depth with the non-spatial logistic density dependent function.
- Non-Spatial Log. Density Dep. Crown Height "a" The "a" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "b" The "b" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "c" The "c" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "d" The "d" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "e" The "e" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "f" The "f" term in the non-spatial logistic density dependent crown depth function.
- Non-Spatial Log. Density Dep. Crown Height "g" The "g" term in the non-spatial logistic density dependent crown depth function.
- Power Function "a" The "a" parameter in the power function for the height-diameter relationship.
- Power Function Exponent "b" The exponent, or "b" parameter, in the power function for the height-diameter relationship.
- Sapling Linear Function Intercept The intercept of the sapling linear function for DBH and height.
- Sapling Linear Function Slope The intercept of the sapling linear function for DBH and height.
- Sapling Reverse Linear Function Intercept The intercept of the sapling reverse linear function for DBH and height.
- Sapling Reverse Linear Function Slope The slope of the sapling reverse linear function for DBH and height.
- Seedling Linear Function Intercept The intercept of the seedling linear function for DBH and height.
- Seedling Linear Function Slope The slope of the seedling linear function for DBH and height.
- Seedling Reverse Linear Function Intercept The intercept of the seedling reverse linear function for DBH and height.
- Seedling Reverse Linear Function Slope The slope of the seedling reverse linear function for DBH and height.
- Slope of Asymptotic Crown Height Slope of the standard equation for determining crown depth.
- Slope of Asymptotic Crown Radius Slope of the standard equation for determining crown radius.
- Slope of Asymptotic Height Exponential decay term in the adult and sapling standard function for DBH and height.
- Slope of DBH to Diameter at 10 cm Relationship The slope of the linear relationsip between the DBH, in cm, and the diameter at 10 cm height, in cm, in small trees. Used by all species.
- Slope of Height-Diameter at 10 cm Relationship The slope of the seedling standard function for diameter at 10 cm and height.
The "standard" crown depth and radius relationships
Crown radius is calculated as:
rad = C1 * DBH a
where:
- rad is the crown radius, in meters
- C1 is the Slope of Asymptotic Crown Radius parameter
- a is the Crown Radius Exponent parameter
- DBH is the tree's DBH, in cm
Crown radius is limited to a maximum of 10 meters.
Crown depth is calculated as
ch = C2 * height b
where
- ch is the distance from the top to the bottom of the crown cylinder, in meters
- C2 is the Slope of Asymptotic Crown Height parameter
- height is the tree's height in meters
- b is the Crown Height Exponent parameter
The "Chapman-Richards" crown depth and radius relationships
The Chapman-Richards equation for calculating crown radius is:
rad = i + a (1 - e -b * DBH) c
where
- rad is the crown radius, in meters
- DBH is the tree's DBH, in cm
- i is the Chapman-Richards Crown Radius Intercept parameter, which represents the crown radius of the smallest possible sapling
- a is the Chapman-Richards Asymptotic Crown Radius parameter
- b is the Chapman-Richards Crown Radius Shape 1 (b) parameter
- c is the Chapman-Richards Crown Radius Shape 2 (c) parameter
The Chapman-Richards equation for calculating crown depth is:
ch = i + a (1 - e -b * H) c
where
- ch is the distance from the top to the bottom of the crown cylinder, in meters
- H is the tree's height, in m
- i is the Chapman-Richards Crown Height Intercept parameter, which represents the crown depth of the smallest possible sapling
- a is the Chapman-Richards Asymptotic Crown Height parameter
- b is the Chapman-Richards Crown Height Shape 1 (b) parameter
- c is the Chapman-Richards Crown Height Shape 2 (c) parameter
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").
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 (radi and chi) 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 * chi 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
- chi 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 chi is as follows:
chi = 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
- radi 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 radi is as follows:
radi = 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
The NCI crown depth and radius relationships
This calculates crown dimensions as a function of tree size and local crowding. The equations are the same for crown depth and crown radius, but they each have separate parameters.
The crown dimensions are calculated as:
CR / CD = [Max CR / Max CD] * Size Effect * Crowding Effect
where:
- CR is the crown radius, in m
- CD is the crown depth, in m
- Max CR is the NCI Crown Radius - Max Potential Radius (m) parameter
- Max CD is the NCI Crown Depth - Max Potential Depth (m) parameter
Size Effect is calculated as:
SE = 1 - exp(-d * DBH)
where:
- SE is the size effect, between 0 and 1
- d is either the NCI Crown Depth - Size Effect "d" parameter or the NCI Crown Radius - Size Effect "d" parameter
- DBH is the tree's DBH, in cm
Crowding Effect is calculated as:
CE = exp(-n * NCI)
where:
- CE is the crowding effect, between 0 and 1
- n is the NCI Crown Radius - Crowding Effect "n" parameter or the NCI Crown Depth - Crowding Effect "n" parameter
- NCI is calculated as below
NCI is calculated as:
where:
- the calculation sums over j = 1...S species and k = 1...N neighbors of each species of at least a DBH of NCI Crown Radius - Minimum Neighbor DBH or NCI Crown Depth Minimum Neighbor DBH, in cm, out to a distance of NCI Crown Radius - Max Search Distance for Neighbors (m) or NCI Crown Radius - Max Search Distance for Neighbors (m)
- α is the NCI Crown Radius - Alpha parameter or the NCI Crown Depth - Alpha parameter
- β is the NCI Crown Radius - Beta parameter or the NCI Crown Depth - Beta parameter
- γ is the NCI Crown Radius - Gamma parameter or the NCI Crown Depth - Gamma parameter
- λik is the NCI Crown Radius Lambda for Species X Neighbors parameter or the NCI Crown Depth Lambda for Species X Neighbors for the target species relative to the kth neighbor's species
- DBHjk is the DBH of the kth neighbor, in cm
- DBHt is the DBH of the target tree for which to calculate crown dimensions, in cm
- distanceik is distance from target to neighbor, in m
DBH - diameter at 10 cm relationship
Seedlings use the diameter at 10 cm as their primary indicator of size, and have no DBH. Saplings use both DBH and diam10. The use of both measurements by saplings helps to maintain continuity between the seedling and adult life history stages. Adults use only DBH.
DBH and diam10 are related as follows:
DBH = (diam10 * R) + I
where
- DBH is the DBH in cm
- diam10 is the diameter at 10 cm height, in cm
- R is the Slope of DBH to Diameter at 10 cm Relationship parameter
- I is the Intercept of DBH to Diameter at 10 cm Relationship parameter
The "standard" diameter-height relationships
"Standard" is one of the names used to describe a set of allometric functions relating height to diameter. There is one for adults and saplings, and one for seedlings. These are called "standard" because they were the original SORTIE functions and until recently were the only choices.
The standard sapling and adult DBH - height function is:
height = 1.35 + (H1 - 1.35)(1 - e-B*DBH)
where:
- height is tree height in meters
- H1 is the Maximum Tree Height, in m parameter
- B is the Slope of Asymptotic Height parameter
- DBH is tree DBH in cm
In some articles, B (Slope of Asymptotic Height) is a published parameter. Other articles instead use H1 and another parameter, H2, which was called the DBH to height relationship. In this case, B can be calculated from published values as B = H2/H1.
The standard seedling diam10 - height function is:
height = 0.1 + 30*(1 - e(-α * diam10))
where:
- height is tree height in meters
- α is the Slope of Height-Diameter at 10 cm Relationship parameter
- diam10 is tree diameter at 10 cm height, in cm
The "linear" diameter-height relationship
The linear diameter-height relationship is the same for all life history stages, but each stage can use a different set of parameter values.
The linear diam - height function is:
height = a + b * diam
where:
- height is tree height, in m
- a is the appropriate linear intercept parameter (either Adult Linear Function Intercept, Sapling Linear Function Intercept, or Seedling Linear Function Intercept)
- b is the appropriate linear slope parameter (either Adult Linear Function Slope, Sapling Linear Function Slope, or Seedling Linear Function Slope)
- diam is DBH (in cm) for saplings and adults, or diam10 (in cm) for seedlings
The "reverse linear" diameter-height relationship
The reverse linear diameter-height relationship is the same for all life history stages, but each stage can use a different set of parameter values. The name comes from the fact that it is almost the same as the linear function, but with height and diameter switched. In other words, in the linear function, height is a linear function of diameter. In the reverse linear function, diameter is a linear function of height.
The reverse linear diam - height function is:
height = (diam - a) / b
where:
- height is tree height, in m
- a is the appropriate reverse linear intercept parameter (either Adult Reverse Linear Function Intercept, Sapling Reverse Linear Function Intercept, or Seedling Reverse Linear Function Intercept)
- b is the appropriate reverse linear slope parameter (either Adult Reverse Linear Function Slope, Sapling Reverse Linear Function Slope, or Seedling Reverse Linear Function Slope)
- diam is DBH (in cm) for saplings and adults, or diam10 (in cm) for seedlings
The "power" diameter-height relationship
The power diameter-height relationship relates height and diameter with a power function. Since it uses diameter at 10 cm, NOT DBH, it is active for saplings only.
The power diam - height function is:
height = a * d10 b
where:
- height is tree height, in m
- a is the Power Function "a" parameter
- b is the Power Function Exponent "b" parameter
- d10 is diameter at 10 cm (in cm)
Last updated: 29-Feb-2008 10:33 AM