Whereas a planted forest, in which the sampling shall be performed by locating not more than just a line of length (L) in the middle of the planting rows, but a cluster with four subunits, each one taken with a distance l of a central point X₀, such that l = 25 m and that the sub-units can be distributed: two following the directions of planting lines and two crossing them at 90˚ right angles, so as to obtain more consistent information in stands with rectangular spacing, as is shown in Figure 1.

As is shown, this unit occupies approximately an area of 2642 m², or little more than 1/4 hectare for the case of fixed area plots and the side L of the fixed area is used for the Strand’s sampling unit.

The cluster’s subunits have remained with the same size proposed by Strand, i.e., L = 15.71 m and the sam- pling were always performed of the left side of the subunits. In these circumstances, it is proposed to identify with a triple indexing, i.e. X_ijk, such that clusters, subunits of the clusters and trees per subunit.

Note that this new proposition requires special considerations for the sample unit, i.e. the estimator of basal area will be performed per subunit and the result per sampling unit will be obtained by the arithmetic mean of these values, once such results will be directly obtained per hectare. In these circumstances, the estimators can remain generically as were developed by Strand, but they should be adapted for the entire cluster unit and the estimator by subunit would be obtained as it is presented in equation (1), held the line length with 5π, i.e. L = 15.71 m:

The estimator of basal area per cluster will be obtained as is presented in (2)

The estimator of basal area average for all clusters will be obtained as is presented in (3)

The estimator of volume will be performed in the same manner and obtained by subunit as is presented in equation (4):

or of volume by cluster will be obtained as is presented in (5):

The estimator of average volume for all clusters will be obtained as is presented in (6):

The volume per sampling unit was calculated for the two sampling methods. In sampling by fixed area indi- vidual volumes for P. elliottii were calculated by means of an equation selected between volumetric models of double entry, having the logarithmic function of Schumacher Hall (7) been selected by Elesbão [9] , from the statistical scores, as the best fit for the species. The same author considered, still, that the models of Spurr, Schu- macher Hall, Spurr logarithmic, BWI—Germany (Forest Research Institute Baden-Wuerttemberg) showed good adjustments, with coefficients of determination greater than 0.80 and lower values relative to the standard error of the estimate. When performing the Strand’s method, it was used a Bitterlich Mirror Relaskop for determina- tion of basal area, followed by the application of a form factor (with reference to 1.3 m) [10] .

where: V = volume (m³); d = diameter at breast height (cm); h = total height (m); b₀, b₁, b₂ = coefficients of model and ln = neperian logarithm.

With the structure of the sample unit as a cluster with four subunits, it becomes possible to consider it suitable to detect the occurrence of dominant trees, as the standard requirement proposed by Assmann [11] .

If in each subunit of the cluster you can have from 6 to 8 trees in the first inventory, with spacing 3 m be- tween rows and 2 m between plants, it counts with at least 28 trees per cluster, i.e., we can have at least a domi- nant tree in each unit, while at a spacing of 3 m × 2 m, of 1666 trees per hectare at least 1 in each 16.66 of them is dominant. Thus, the selection of a dominant tree is perfectly assured in this sample structure.

Strand in his work had not presented the solution to obtain the average height of the sample unit, since the me- thod does not presuppose to measure individual heights [1] . The sampling is conducted with selection propor- tional to the height of the trees, using the scientific formulation proposed by Lorey.

With two approaches used in the same unit you can avail this condition to reach a consistent estimator of av- erage height of Lorey.

Consider the estimator of volume obtained by Strand, presented in (4), i.e., after the inclusion of cluster unit, however with indexing of the diameters for this first estimation method, i.e., as is shown in (8):

Consider, in addition that, if the estimator is developed for the basal area in (1), we are also able to obtain the volume of trees per hectare, qualified in the sampling methodology indexed as 2, as is shown in (9).

where is the average height of Lorey

Assuming that both estimators are consistent and provide results so similar, such that if you can accept them as equals, i.e., then equating the estimators (8) and (9) you get:

and the average height of Lorey can be obtained easily as follows

This method applied for the first time in Brazil by Figueiredo [12] estimates the volume based on the methodol- ogy of Hohenadl [13] , cited by Prodan [14] , which divides the bole in n parts of equal relative sizes as a function of the total height of the trees. However, according to Prodan [14] , the smaller is the length of sections closest to the exact will be the estimated value of the volume. In this way, it was decided in this study divide the bole in ten sections of relative length, using the methodology of Hohenadl, as shown in Figure 2.

With the aim of determining the relations between natural and artificial form factor, Ko introduced the fol- lowing concept about the mean quadratic diameter [15] :

Consider the form factor calculated by Hohenadl:

or:

In that the sum of the square of the diameters of the series, divided by the number of these, provides the mean quadratic diameter, i.e.:

The Hohenadl’s form factor is then obtained as follows:

This concept of form factor has been widespread as is presented in expression (15), which uses a referential diameter used to obtain any desired form factor, assuming that the ten diameters along the bole are taken in rela- tive way:

In that: d_x = reference diameter and d_0.i = measured diameters in the relative sections.

Taking the mean quadratic diameter as referential diameter, you can obtain the unitary form factor [14] .

The form of the tree, in this case, is equivalent to the cylinder, whose diameter is coincident with the square root of the mean quadratic diameter. In this way, the volume is obtained as follows:

Given the difficulty to find out the location of d_q in the boles and also to measure it, we tried to establish a re- lationship between the mean quadratic diameter with the dbh, which is a variable of easier access and mea-

surement, using regression equations, as proposed by Ko (1968), i.e. using parables.

where: a, b and c are the coefficients of the parabolic models.

Replacing the result obtained in (18) or (19) into (17) we can obtain the volume of trees, as suggested by Ko [15] . In work done by Péllico Netto, the best results for the species Araucaria angustifolia were achieved with the equation (19) [16] .

The adjustments were performed using the procedure of linear regression applying the least square method and the parameters were statistically evaluated by t-test at the level of 95% of probability. To check the reliabil- ity of these adjusted models, the statistics: standard error (s_yx%) and adjusted coefficient of determination (R²) were also obtained.

In these circumstances, the volume will be estimated by subunit directly per ha (20), as shown in Equation (8):

The estimator of volume by cluster will be obtained as is presented in (21).

The estimator of average volume for all clusters will be obtained as is presented in (22)

Equating, as shown in (19) and replacing the results of the function in (23), (24) and (25), the volumetric estimator for each subunit can be obtained directly as a function of dbh, i.e.:

The estimator of volume by cluster will be obtained as is presented in (24)

The estimator of average volume for all clusters will be obtained as is presented in (25).

Under the proposed conditions, the estimators will only be obtained if the equation, as shown in (19), is de- veloped previously for the area to be sampled where it will be applied this methodology.

To obtain volume, it was measured initially the ten bole diameters in each of the trees sampled and calculated their quadratic diameters and these were used to develop the linear regression as a function of dbh, as suggested in (19).

The adjustment resulted in coefficient of determination equal to 0.99 and standard error of the estimate of 5.67%, from the indicated function in (26), and the residuals are presented in Figure 3.

The volumetric calculations for trees, by subunits and by cluster were performed as indicated in (22), (23), (24) and (25).

The results for both approaches are presented in table 2.

It can be observed that both methodologies showed similar results. The estimated mean volume using the methodology of fixed area was 3.4% higher than that calculated by the method of Strand. Referring to the average basal area, both methodologies showed very close values, being the result obtained by the methodology with fixed area 0.06% lower than that obtained using the methodology of Strand (36.143 m²∙ha⁻¹ and 36.166 m²∙ha⁻¹ respectively) (Table 2).

By comparative analysis considering a t-test for independent samples, no statistical differences were found between the means of 10 clusters, referring to basal area (t_calc= 0.00046 < t_18;0.025 = 2.44) and volume (t_calc= 0.02453 < t_18;0.025 = 2.44).

Consequently, through the analysis of the results, it is concluded that the pps sampling as proposed by Strand, adapted to the system of clusters proved to be efficient, generating results very close to those obtained with the sample by fixed area, thus consolidating the next steps of this study.

Considering each cluster, equations (8) and (9) were assumed to be equivalent to determine the volume. There- fore, by substituting the estimators of volume, the values for the height of Lorey were determined indirectly, us- ing equation (11), considering only the values of diameter and quadratic mean diameter. The values calculated by the indirect method were compared with those obtained in fixed area sampling, as indicated in Table 3.

Both estimates were very similar, with equal means for clusters using the two methodologies. The proximity between the heights was also verified statistically and no differences between the means were detected for the two methods (t_calc = 0.04 < t_18;0.025 = 2.44). Therefore, the calculation of indirect height of Lorey from diameters showed to be feasible and efficient, whose estimates were consistent. The process of indirect calculation allows for greater efficiency in the sampling process, making unnecessary the direct measurement of heights.

The methodology proposed is among the most attractive solutions to reduce the cost of inventories in planted forests, with the ability to have greater accuracy and greater scope for information at the level of compartments, without increasing the cost of sampling performed with units of fixed area.

The use of smaller permanent sampling units with higher intensity in the compartments before the final cut will substantially increase the precision of the estimators in these management units, which will enable to elimi- nate the pre-cut inventory in forest enterprises.