Forest Mensuration. Brack and Wood


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Measurement
Angle count
Practical hints
Related measurements

Stand Basal Area ©


Stand Basal Area is a useful parameter for the forest mensurationist because it is relatively easily collected and can be related to many other parameters of interest (e.g. site density and stand volume). The internationally accepted symbol for stand basal area is G (m^2 ha^-1) and for tree basal area, g (m^2). G is an important reference variable for a stand and is particularly useful in quantitative description.

G values commonly range from 10 to 60 m^2 ha^-1 in both coniferous and hardwood forests. In rare cases, G values of 150 m^2 ha^-1 may be reached on exceptionally good sites.

G is the sum of the basal areas of all (living) trees, in the stand. G is derived by reference to tables or by summation by calculator:

G (square metres) = 0.000 078 539 8 x d^2, where d is DBH in cm.

Note: 0.000 078 539 8 is (PI/40 000). The division by 40 000 corrects for the difference in units (cm and m) and diameter to radius.

G is usually expressed over bark and on a unit area basis, e.g. m^2 / ha.

Stand mean BA is also a useful stand variable. It is calculated as G/N, where N is the number of trees in the stand. The diameter equivalent to g is called the QUADRATIC MEAN DBH:
dg = sqrt( (G / N) x 40000 / PI)
where dg is the international symbol for quadratic mean DBH.

The quadratic mean DBH is preferable to the arithmetic mean DBH as a size parameter of trees in a stand because of the additional weight it gives to the larger diameters.

Estimating stand basal area

The basal area of a forest stand is estimated by sampling in one of two ways:-

Fixed area plots

The DBH of each tree is measured using circular, rectangular or square sample units (plots) and mean stand basal area (G) is determined by totalling the basal area of each tree in the plot, and dividing by the area of the plot. Sampling using fixed area plots is not as efficient as angle count sampling for estimating G.

Angle count sampling

When a population mean depends more on the size of large compared with small units, it is more efficient to select the larger units with greater probability. This technique is called PPS Sampling or sampling with Probability Proportional to Size.

This principle has been applied to various forest sampling problems but most notably to estimating basal area per unit area by angle counting, where inclusion of a tree in the count depends on the basal area of the tree and its proximity to the sampling point, ie small trees are not included if they are some distance from the sampling point, while larger trees will be included at even greater distances.

Angle count sampling estimates G per unit area of a stand without measuring:

The method was developed by an Austrian forester Dr W. Bitterlich in the late 1930s and perfected in the late 1940s. Its application was recognised by Cromer (1952) in Australia and Grosenbaugh (1952) in the USA ahead of European foresters. In the literature, it is referred to under various names, namely:

Angle count sampling procedure

Principle of angle count method

A simple "mind experiment" might help explain the principle of the method:

Imagine that there exists a forest with only small and large diameter trees (e.g. 10 cm and 50 cm DBH respectively) for which we want to determine stand basal area. A single 10 cm DBH tree only has a basal area of 0.00785 m^2 while each 50 cm tree is 0.196 m^2 (= PI * Radius^2 ). We do not want to waste time measuring too many small trees, but do not want to miss the big values contributed by the large trees, so we use 2 circular plots of 5 m and 25 m radius and measure only small and large trees respectively within each plot. Diagram of concentric plots

Now imagine that we established our plots and found that there were 3 small trees within the 5 m radius plot and 4 large trees within the 25 m radius plot. The stand basal area would be calculated as:

10 cm tree contribution:        
 = 3 * (PI * Radius(tree)^2 ) / (PI * Radius(plot)^2) (m^2/m^2)
 = 3 * (PI * 0.05^2) / (PI * 5^2) (m^2/m^2)
 = 3 * 0.00785 / 78.5 (m^2/m^2)
 = 3 * 0.00001 (m^2/m^2)
 = 3 * 0.00001 * 10000 (m^2/ha)
 = 3 * 1 (m^2/ha)


50 cm tree contribution:
 = 4 * (PI * 0.25^2) / (PI * 25^2)  (m^2/m^2)
 = 4 * 0.196 / 1960 (m^2/m^2)
 = 4 * 0.00001 (m^2/m^2)
 = 4 * 0.00001 * 10000 (m^2/ha)
 = 4 * 1 (m^2/ha)


Stand basal area        = 3 + 4 (m2/ha)  = 7 (m2/ha)

But of course a real forest would have trees of a range of DBH values. We could therefore establish a range of plots for all the different DBH classes. For example, our imaginary forest above might also have trees of 20, 30 and 40 cm DBH which we could sample in circular plots of 10, 15 and 20 m radius respectively. Using the same calculations as above therefore, we would find that each 20, 30 or 40 cm DBH tree that is within its 10, 15 or 20 m radius plot adds 1 m^2/ha to the overall stand basal area. Now that is the important part of the matter! We have set up our plot dimensions so that each tree of X cm DBH adds 1 m^2/ha to the stand basal area if it is within a radius X/2 m of the plot centre - where X is any number. In fact, we no longer even need to know what is the value of X. All we need to know is whether the ratio of DBH to distance from the plot centre is greater than 2 cm : 1 m. If the ratio of DBH : Distance is greater than 2 cm : 1 m (1 : 500), then the tree of whatever DBH is within its respective plot and that tree adds another 1 m^2/ha to the estimate of the total stand basal area. An optical wedge, Dendrometer II, Spiegal Relakop or similar instrument simply helps determine if the ratio of DBH : Distance away exceeds the critical ratio and therefore whether the tree is within the plot and adds to the estimate of stand basal area.

Note that a tree of X cm DBH is within its respective plot when the ratio of DBH : Distance from centre exceeds 2 * X cm : X m. Thus a tree of 20 cm DBH is counted if it is anywhere from 10 m away right up to the exact plot centre! A tree is either in or out of the plot - a 20 cm tree right at the centre is no more or less within the plot than a similar tree that is 3, 6 or 9 m away from the centre.

Alternatively, the following argument may help explain the pinciples:
Assume:
Diagram of angles

Now:

The value 10 000 Sin^2Q, (which is the equivalent to 10 000 x R^2 / Y^2) is known as the basal area factor or BAF. Stand basal area = N x BAF.

Angle count instruments

Collectively, instruments used in angle count sampling procedures are termed angle gauges. Built into each gauge is a certain reference angle. When using an angle gauge, this reference angle is compared with the angle subtended at a fixed point (the angle count spot) by the sides of a tree (usually at breast height). If the subtended angle is larger than the reference angle, then the tree is included in the angle count.

Instruments include:

Practical aspects when using angle count instruments

  • If a 360 sweep is not possible (e.g. boundary of stand) accept 180 or 90 sweeps when necessary and weight the estimate accordingly, i.e. x 2 or x 4 (Grosenbaugh's method). This eliminates bias. Alternatively, one can apply what is called the "mirage" method (Schmid-Haas's method). For sample points near the forest margin, make a 360 sweep as usual. Then measure the horizontal distance from the angle count spot perpendicular to the forest margin. Extend the line an equal horizontal distance into the area beyond the margin and locate the "mirage" angle count spot. Then, from this spot, make another 360 sweep. Stand density at the point in the forest is then derived by summing the two estimates.
  • Bias is also likely with LEANING or ECCENTRIC stems. The latter is the more serious and nothing can be done about it. One hopes that in a full sweep, the errors will compensate. With leaning stems, align the angle gauge at right angles to the leaning axis of the stem. Be careful with trees which lean towards or away from the observer, particularly if they are borderline. In this case, the check distance is to the centre of the stump.
  • When one tree is obscured by another, move sideways on the radius, i.e. keep distance from tree constant. Then make the reading and return to the angle count spot.
  • Be alert for dead trees which normally would be excluded from assessment. If a stand is composed of several species, separate basal area estimates can be made sfor each species by keeping a separate tally.
  • Establishing angle count spots

    Disadvantages of Angle Count Sampling

    Some Experiences with Angle Count Sampling

    Van Laar (S. Afr. J. For. 72: 1-6 (1970)) estimated G in 77 sample plots using the Spiegel Relaskop, a wedge prism and calipers. He found:

    Whyte and Tennent (N.Z. J. For. 20 (1): 134-47, 1975) point out that whenever angle count and bounded plot estimates of mean basal area per unit of area are compared, foresters almost invariably assume that the bounded plot gives the correct value, and this is then used to judge the accuracy of the angle count estimate. This assertion is quite wrong since a theoretically unbiased estimate of mean basal area should sample in proportion to stem basal area, not stem frequency. Bounded plot sampling does not achieve this whereas angle count sampling does. Both samples, however, provide only estimates of the true population mean. Each is subject to sampling error.

    Publications by Palley and O'Reagan (For. Sci. 7: 282-93, 1961) and Kulow (J. For. 64: 469-74, 1966) also attest to the superiority of angle counting in accuracy and precision for estimating mean stand basal area. However, all authors agree that bounded plots are more efficient for determining stocking density (number of stems per unit area).

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    http://online.anu.edu.au/Forestry/mensuration/S_BA.HTM
    Cris.Brack@anu.edu.au
    Tue, 1 Nov. 1997