The Small-tree Height Increment Model

For smaller sized trees it is difficult to sample for periodic diameter increment. There may be less than 10 years’ growth at breast height, and removal of an increment core

could severely damage the tree. However, with many species counting whorls and measuring internodes can estimate height increments for small trees for periods up to 5 years. This is a common inventory procedure.

Height growth for small trees is a driving force in tree development as they compete for light and vertical growing space. Because of this, the small-tree portion of the Forest Vegetation Simulator is a height growth driven model. That is to say, for small trees less than some threshold diameter (see section 6.5.1) height growth is estimated first, and diameter growth is predicted from height growth.

Because the small-tree height increments can be measured with relative ease, and because the height growth equation is so influential in the development of small trees, FVS

includes a calibration procedure for the small-tree height increment model. This

procedure is analogous to the procedure used in the large-tree diameter increment model. The median residual between observed and predicted height increments is computed on the logarithmic scale and incorporated in the prediction equation as an additional intercept term.

Equations used to predict small-tree height increment vary by species and variant.

However, they are usually dependent on factors such as site characteristics, stand density, social position, and crown ratio. The variant overview document contains equations and coefficients particular to the variant being used.

The difficulty with this approach centers on the choice of a break point diameter at which the switch is made from a height growth driven small-tree model to a diameter growth driven large-tree model. This is best illustrated by discussing experiences developing the North Idaho variant. Stage (1975) developed a periodic height increment model based on the differential of the allometric relationship between height (HT) and diameter (DBH). Periodic (10-year) height increment (HTG1) is predicted as a function of HT, DBH, 10-

year DBH increment (DG), species, and habitat type.

A series of modifications were implemented in the basic model to force height increment to slow down in very tall trees even though diameter increment may still be quite

substantial. In the modified form shown in equation {7.2.2.1}, coefficients of the DG and HT2 terms are dependent on habitat type and coefficients of the DBH term are dependent on species.

{7.2.2.1} ln(HTG1) = HAB + SPP + b1 · ln(HT) + b2 · ln(DBH) + b3 · ln(DG) +

b4 · HT2

where:

HAB is a constant dependant on habitat type SPP is a constant dependant on species

b1-b4 are regression slope coefficients dependent on species and/or habitat

In Stage’s height increment model, many of the effects related to site characteristics and stand conditions are indirectly represented in the diameter increment term. In addition,

Stage’s model was applicable to all sizes of trees, and is used for large-tree height growth estimation in the North Idaho variant.

Subsequently, the independent model shown in equation {7.2.2.2} was developed

Wykoff et al. (1982) to predict periodic (5-year) height increment (HTG2) for small trees.

This model has explicit site and stand density variables and no diameter increment term. {7.2.2.2} ln(HTG2) = LOC + HAB + SPP + b1 · ln(HT) + b2 · CCF + b3 · SL ·

cos(ASP) + b4 · SL · sin(ASP) + b5 · SL

where:

HAB is a constant dependant on habitat type, plant association, or site index LOC is a constant dependent on location code (National Forest)

SPP is a constant dependant on species ASP is stand aspect

SL is stand slope ratio (percent / 100) CCF is stand crown competition factor

b1-b5 are regression slope coefficients dependent on species and/or location

With two independent models to predict the same attribute (height growth of the small trees), they were unable to find a suitable tree size for transition between models.

Regardless of the diameter chosen as a breakpoint, a discontinuity in the response surface existed.

This problem was resolved by using a simple switching function. For trees with DBH less than a threshold diameter, the height increment prediction is based entirely on the small- tree model; for trees with DBH greater than a second threshold diameter, the prediction is based entirely on the large-tree model. If DBH is between the two threshold diameters, the two estimates are combined using equations {7.2.2.3} and {7.2.2.4}, with the large- tree prediction (HTG1) being given weight of HWT, and the small-tree prediction (HTG2)

being given a weight of (l-HWT).

{7.2.2.3} HTG = HWT · HTG1 + (1 - HWT) · HTG2

{7.2.2.4} HWT = (treeDBH - lowerDBH)/(upperDBH - lowerDBH) where:

lowerDBH is the lower threshold diameter upperDBH is the upper threshold diameter treeDBH is the tree’s diameter at breast height

Examining the composite behavior of the model reveals that the height increment curve increases rapidly to a maximum at 3 to 5 inches DBH and then gradually decreases, much in the fashion of the classical increment curve (Assmann 1970). The effect of increasing density is to suppress height increment—directly through the CCF term in the small-tree model, indirectly through the DG term in the large-tree model.

In an undisturbed even-aged stand, the height and diameter increment models work together to produce increasingly flattened height-diameter curves over time.