Allometry
In this document:
Allometry parameters
The "standard" crown height and width relationships
The "Chapman-Richards" crown height and width 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.
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 Crown Height-Height Function The allometric relationship used to relate tree height (in m) and height (or length) of the crown (in m) for adult trees. Each of the choices is described in detail below.
- Adult Crown Radius-Diameter Function The allometric relationship used to relate DBH (in cm) and crown radius (in m) for adult trees. Each of the choices is described in detail below.
- Adult Height-Diameter Function The allometric relationship used to relate DBH and height for adult trees. Each of the choices is described in detail below.
- Adult Linear Function Intercept The intercept of the adult linear function for DBH and height. Any species not using that function can omit this value.
- Adult Linear Function Slope The slope of the adult linear function for DBH and height. Any species not using that function can omit this value.
- Adult Reverse Linear Function Intercept The intercept of the adult reverse linear function for DBH and height. Any species not using that function can omit this value.
- Adult Reverse Linear Function Slope The slope of the adult reverse linear function for DBH and height. Any species not using that function can omit this value.
- Chapman-Richards Asymptotic Crown Height The asymptotic crown height (or length), in m, of the Chapman-Richards crown height equation. Any species not using that function can omit this value.
- Chapman-Richards Asymptotic Crown Radius The asymptotic crown radius, in m, of the Chapman-Richards crown radius equation. Any species not using that function can omit this value.
- Chapman-Richards Crown Height Intercept The intercept of the Chapman-Richards crown height equation. This represents the crown height, in m, of the smallest possible sapling. Any species not using that function can omit this value.
- Chapman-Richards Crown Height Shape 1 (b) The first shape parameter, b, of the Chapman-Richards crown height equation. Any species not using that function can omit this value.
- Chapman-Richards Crown Height Shape 2 (c) The second shape parameter, c, of the Chapman-Richards crown height equation. Any species not using that function can omit this value.
- 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. Any species not using that function can omit this value.
- Chapman-Richards Crown Radius Shape 1 (b) The first shape parameter, b, of the Chapman-Richards crown radius equation. Any species not using that function can omit this value.
- Chapman-Richards Crown Radius Shape 2 (c) The second shape parameter, c, of the Chapman-Richards crown radius equation. Any species not using that function can omit this value.
- Crown Height Exponent The exponent in the standard equation for calculating crown height. Any species not using that function can omit this value.
- Crown Radius Exponent The exponent in the standard equation for determining the crown radius. Any species not using that function can omit this value.
- 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.
- Power Function "a" The "a" parameter in the power function for the height-diameter relationship. Any species not using that function can omit this value.
- Power Function Exponent "b" The exponent, or "b" parameter, in the power function for the height-diameter relationship. Any species not using that function can omit this value.
- Sapling Crown Height-Height Function The allometric relationship used to relate tree height (in m) and height (or length) of the crown (in m) for saplings. Each of the choices is described in detail below.
- Sapling Crown Radius-Diameter Function The allometric relationship used to relate DBH (in cm) and crown radius (in m) for saplings. Each of the choices is described in detail below.
- Sapling Height-Diameter Function The allometric relationship used to relate DBH and height for saplings. Each of the choices is described in detail below.
- Sapling Linear Function Intercept The intercept of the sapling linear function for DBH and height. Any species not using that function can omit this value.
- Sapling Linear Function Slope The intercept of the sapling linear function for DBH and height. Any species not using that function can omit this value.
- Sapling Reverse Linear Function Intercept The intercept of the sapling reverse linear function for DBH and height. Any species not using that function can omit this value.
- Sapling Reverse Linear Function Slope The slope of the sapling reverse linear function for DBH and height. Any species not using that function can omit this value.
- Seedling Height-Diameter Function The allometric relationship used to relate diameter at 10 cm and height for seedlings. Each of the choices is described in detail below.
- Seedling Linear Function Intercept The intercept of the seedling linear function for DBH and height. Any species not using that function can omit this value.
- Seedling Linear Function Slope The slope of the seedling linear function for DBH and height. Any species not using that function can omit this value.
- Seedling Reverse Linear Function Intercept The intercept of the seedling reverse linear function for DBH and height. Any species not using that function can omit this value.
- Seedling Reverse Linear Function Slope The slope of the seedling reverse linear function for DBH and height. Any species not using that function can omit this value.
- Slope of Asymptotic Crown Height Slope of the standard equation for determining crown height. Any species not using that function can omit this value.
- Slope of Asymptotic Crown Radius Slope of the standard equation for determining crown radius. Any species not using that function can omit this value.
- Slope of Asymptotic Height Exponential decay term in the adult and sapling standard function for DBH and height. Any species not using that function can omit this value.
- 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. Any species not using that function can omit this value.
The "standard" crown height and width 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 height 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 height and width 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 height 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 height 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
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: 25-Jan-2006 03:17 PM