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

The "standard" crown depth and radius relationships

Crown radius is calculated as:

rad = C1 * DBH a

where:

Crown radius is limited to a maximum of 10 meters.

Crown depth is calculated as

ch = C2 * height b

where

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

The Chapman-Richards equation for calculating crown depth is:

ch = i + a (1 - e -b * H) c

where

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:

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:

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

Non spatial logistic density dependent crown depth function

where:

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:

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:

Size Effect is calculated as:

SE = 1 - exp(-d * DBH)

where:

Crowding Effect is calculated as:

CE = exp(-n * NCI)

where:

NCI is calculated as:

where:

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

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:

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:

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:

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:

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:


Last updated: 29-Feb-2008 10:33 AM