Ishimori equation

Stand density index (also known as Reineke's Stand Density Index after its founder) is a measure of the stocking of a stand of trees based on the number of trees per unit area and diameter at breast height of the tree of average basal area. It may also be defined as the degree of crowding within stocked areas, using various growing space ratios based on crown length or diameter, tree height or diameter, and spacing. Stand density index is usually well correlated with stand volume and growth, and several variable-density yield tables have been created using it. Basal area, however, is usually satisfactory as a measure of stand density index and because it is easier to calculate it is usually preferred over SDI. Stand density index is also the basis for Stand density management diagrams.

Calculating Stand Density Index

Plotting the logarithm of the number of trees per acre against the logarithm of the quadratic mean diameter (or the dbh of the tree of average basal area) of maximally stocked stands generally resulted in a straight-line relationship. In most cases the line was used to define the limit of maximum stocking. This negatively sloped line, is referred to as the self-thinning line or the maximum-density line.

The maximum density line is expressed by the equation: log₁₀N = -1.605(log₁₀D) + k

Where N = number of trees per acre D = dbh of the tree of average basal area k = a constant varying with the species

When the quadratic mean diameter equals Template:Convert, the log of N equals the Stand Density Index.

In equation form: log₁₀SDI = -1.605(1) + k

Which means that: k = log₁₀SDI + 1.605

Substituting the value of k above into the reference-curve formula gives the equation:

log₁₀N = log₁₀SDI + 1.605 - 1.605 log₁₀D

This equation can be rewritten as:

log₁₀SDI = log₁₀N + 1.605 log₁₀D - 1.605

The above equation is an expression for computing the stand density index from the number of trees per acre and the diameter of the tree of average basal area.

Let's assume that a stand with basal area of Template:Convert and 400 trees per acre is measured. The dbh of the tree of average basal area D is:

${\sqrt {\frac {150}{400\times .005454}}}=8.29$

Substituting this value into the stand density equation gives:

log₁₀SDI = log₁₀(400) + 1.605log₁₀(8.29) - 1.605 = 2.47

SDI = 10^2.47

SDI = 295

The computed value of SDI is often compared to the species maximum to determine the relative "stand density" or stocking of the stand.

References

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