Use data from permanent sampling points in growth and yield modeling

This study was carried to evaluate the efficiency of the Bitterlich method in growth and yield modeling of the even-aged Eucalyptus stands. 25 plots were setup in Eucalyptus grandis cropped under a high bole system in the Central Western Region of Minas Gerais, Brazil. The sampling points were setup in the center of each plot. The data of four annual mesurements were colleted and used to adjust the three model types using the age, the site index and the basal area as independent variables. The growths models were fitted for volume and mass of trees. The efficiency of the Bitterlich method was confirmed for generating the data for growth and yield modeling.


INTRODUCTION
The choice of the growth and yield model type depends greatly on the management objectives. According to Campos and Leite (2002), it is as important to define the suitable model for a particular situation to know the data characteristics necessary for its construction, as well as the sampling method and their consequences for the efficiency of the model.
The main data sources for modeling are: permanent plots set up for the specific purpose of modeling, permanent plots from the continuous forest inventory (PPCFI), permanent plots set up solely to obtain data to monitor chances in the object population (specific experimental design) and partial stem analysis. The use PPCFI prevails in Brazil.
The PPCFI plot size varies depending on stand heterogeneity and management type. Rectangular plots of about 600 m 2 are commonly adopted in Brazil for unthinned Eucalyptus and Pinus plantations. It is also common the use of rectangular plots of approximately 1000 m 2 in thinned plantations of Eucalyptus Pinus and Tectona. These fixed-area plots are generally set up randomly or in some cases, selectively (deliberate location of the plots in the stands).
The Bitterlich method was proposed many years ago, however, its use was and has been disregarded by many forestry engineers over the years in Brazil and in other countries. There are many doubts about its efficiency for inventory purposes and growth and yield modeling. Among the first studies reported on this method were those by Grosenbaugh (1952), Avery (1955), Hirata (1955), Strand (1957, Grosenbaugh and Stover (1957), Kirby (1965), Bell and Alexander (1967). A list of publications on this method, from 1959 to 1965, was prepared by Thomson and Deitschman (sd).
This study presents the results of a study carried out to assess the efficiency of permanent sample points for even-aged growth and yield modeling.

MATERIAL AND METHODS
Twenty five permanent rectangular plots of 300 m 2 were established in a Eucalyptus grandis stand with initial 3.0 x 2.0 m spacing, in a high bole system, located in the Central Western Region of Minas Gerais, Brazil. These plots were measured at 34, 46, 57 and 73 months. A pole measuring about 1.3 m above soil level was placed in the center of each fixed-area plot. A hole was left on the upper part of this pole large enough to attach a support for a standard relaskop.
The trees classified in the first measurement of each plot using the basal factor 1 of the standard relaskop were numbered from 1 to n. In the subsequent measurements, the new classified trees were also numbered from n+1 to p sequentially.
Diameters of all the trees, in each type of plot, were measured. In the fixed-area plots, the total height of the first seven trees was measured, as well as the total height of the three dominant trees identified among the trees classified with the factor 1. These measurements were taken in four occasions. Furthermore, six trees in each diameter class were felled and cubed, with width of 2.0 cm at the last measurement, resulting in 98 sample trees. Wood discs were removed from each sample tree at the positions 0%, 25%, 50%, 75% and 100% of the commercial height (height with a diameter of approximately 4.0 cm) and were used to determine the wood density of each sample tree. The density per sample tree was obtained using the methodology by Campos and Leite (2002).
The obtained cubing data were used to adjust the multiple volumetric model proposed by Leite et al. (1995) as follows: (1), where V i = volume per sample tree, up to the commercial diameter d, in m 3 ; dbh = diameter at breast height outside bark in centimeters; Ht = total tree height, in m; e = neperian logarithm base Tx = dummy variable (0 for V i outside bark and Tx = 1 for V i inside bark); β 1 = regression parameters i = 0, 1, ..., 4; and; ε = random error, with .
The discs containing wood and bark, from the 98 sample trees, were used to determine wood and bark densities. The mass for each sample tree was obtained from the densities and the volume up to the commercial diameter d. The data were used to adjust the multiple mass model:

Use of data from permanent sampling points in …
(2), where W = tree mass up to the commercial diameter d and Tx = 0 for mass outside bark and 1 for mass inside bark.
The model LnHt = β 0 + β 1 /dbh + β 2 LnHd + ε (3) was fitted with data of each measurement from the 25 permanent fixed-and variable-area plots, where Hd is the mean dominant height of the plot. The equations obtained were used to estimate the trees that had only the dbh measurement in each type of plot.
The equations obtained by the fit of the models (1), and (2), and (3) were used to process the fixedarea and variable-area plots.
The volume and basal area estimates obtained for each year, with fixed and variable-area plots, were compared using the procedure proposed by Leite and Oliveira (2002). A synthesis of this procedure is presented in the Appendix.
To obtain the site index of each plot, the dominant height and age data were used to fit the Richards model as follows: (4) where Hd is the mean dominant height of the plot in meter, and I is the age, in months.
Specific site index equations were not generated for variable area methodology because the variablearea plots were in the same locations as the fixed-area plots, resulting in the same guide-curve. The age index adopted was 60 months.
To estimate growth and yield for each type of plot, the models including the variables I, I and S, and I, S and B were fitted. The models were adjusted for volume (V) and mass (W) with bark up to a minimum commercial diameter of 4.0 cm using the Quasi Newton procedure, except for the Clutter model , V=f (I,S,B), fitted by the two stages least squared method. The models fitted were where: Y 2 = volume, m 3 ha -1 or mass, kgha -1 , at the future age I 2 ; I 1 and I 2 = current and future ages, in months; Growth and yield curves were constructed with the equations obtained from the fixed and variablearea plot data and the technical rotation was determined for each case. The estimates of yield in wood volume and mass obtained with the two data sources were compared using the procedure proposed by Leite and Oliveira (2002).

RESULTS AND DISCUSSION
The hypsometric equations obtained with fixed and variable-area plot data did not differ statistically by the F test proposed by Graybill (1976). Thus, equations common to the two types of plots were used for each age (Table 1). The multiple volume and multiple mass equations presented below, along with the hypsometric equations (Table 1), were used to process the plots at all ages (Table 2). The total height and quadratic diameter measurements obtained with the fixed-and variablearea plot were significantly different (p < 0.05) by the statistical procedure proposed by Leite and Oliveira (2002) (Table 3). Similar results were reported by Couto et al. (1993) in a temporary inventory conducted in a four-year-old Eucalyptus saligna stand. Basal area statistics were statistically equal after 46 months, by the same statistical procedure (p > 0.05). Except for the first volume measurement and the second mass measurement, the wood volume and mass measurements obtained with the two types of plots were statistically equal (p > 0.05).
Generally, it can be inferred that both types of plots can be used to estimate basal area, wood volume and mass at consecutive ages, that is, for continuous forest inventory. Although the precision of the inventory was not assessed, when interpreting the confidence intervals for wood volume and mass, implicit in Table  2, it was found that both intervals contained the means obtained with the two types of plots, at 46, 57 and 73 months.

Growth and yield modeling
The equation obtained to estimate the dominant height with determination coefficient (R 2 ) equal to 0.736 was: Considering the functional relationship above and that the site index is the dominant height at the index age (60 months), it results in: These expressions were used to generate the anamorphic site index curves ( Figure 1) and the site indexes per plot. Table 4 shows the growth and yield models fitted with the fixed-area and variable-area plot data.
Residual analyses do not indicate bias. However, more precise estimates were obtained with the V=f(I,S,B) model type, as expected. This type of model is frequently used for eucalyptus and pine plantations in Brazil. Table  4, the technical rotations were determined for the site indexes 28, 22 and 16, considering the mean initial basal areas observed at 34 months. The rotation did not differ greatly when one or the other plot type was adopted (Table 5). This indicates that the variable-area plots can in principle be used in growth and yield modeling. The rotations were obtained by -β 1 S -1 for the models V=f(I,S) and W=f(I,S); the models V=f(I) and W=f(I) were those corresponding to the maximum mean monthly increase. Except for model W=f (I,S,B), the rotations indicated in Yj and Yi indicate estimates obtained with variable-and fixed-area plots, respectively. ns = p> 0.05 and * = p< 0.05. The model LnB 34 =β 0 +β 1 S+ε was fitted to simulate growth and yield using the Clutter model, V=f (I,S,B) and W=f(I,S,B) from different site indexes and initial basal areas, using data from the first measurement of the permanent plots, resulting in LnB 34 = -0,068004289 + 0,088872711(S) with r 2 = 0.691 and B 34 , the basal area at 34 months. Although statistically equal (Table 6), the difference among volume estimates increase with the decrease in the site index ( Figure 2). The yield curves are coincident from the 24 m site index. The greatest differences detected when the W=f(I,S,B) model was adopted may have occurred because of the worse fit found for this model when variable-area plots were adopted to estimate mass (Table 4).

Using the equations V=f(I,S,B) and W=f(I,S,B),
The statistical procedure proposed by Leite and Oliveira (2002) was also used to compare the models generated with data from fixed and variable area plots. For this purpose, volume and mass estimates were generated for the site index and basal area recorded at the first measurement (34 months). Wood volume and mass were projected from these data for the ages

Model
Type of plot Equation   Figure 3). Therefore, it can be inferred that permanent sampling point can be used for growth and yield modeling.   Mass (t.ha -1 ) * * yes Y j ≠ Y 1 V = f(I,S) Volume (m 3 ha -1 ) ns ns yes Y j = Y 1 V = f(I,S) Mass (t.ha -1 ) ns ns yes Y j = Y 1 V = f(I) Volume (m 3 ha -1 ) ns ns yes Y j = Y 1 V = f(I) Mass (t.ha -1 ) ns ns yes Y j = Y 1 ns = p>0.05 and * = p< 0.05.

CONCLUSIONS
Based on the results it can be concluded: -permanent sampling point are efficient to conduct continuous forest inventories; -data obtained from permanent sampling points are efficient for growth and yield modeling at stand level; -the Bitterlich method enables precise and umbiased estimates of basal area, volume and mass per hectare in eucalyptus plantations, at different ages; Although it was not the objective to study regular mortality, it can be inferred that the use of permanent sampling points does not hinder its quantification. Quantification can be carried out between any two ages if only the trees classified on the first of these two occasions is considered.