^{1}

^{*}

^{1}

^{1}

The forest management, in broad terms, covers a set of information and activities related to the growth and production of plantations or from heterogeneous forests. If on one hand forest plantation presents homogeneous dendrometric characteristics, on the other hand in heterogeneous forests such a configuration does not exist, which requires the application of essential mathematical estimates for the forest management planning, for example, the prognosis of diametric structure. In this respect, refinements can be tested in traditionally used mathematical models, aiming at more accurate results. These facts substantiate the following hypotheses: the stratification of the diameter increments can provide estimates with greater accuracy and more precise results for the prognosis. With the stratification of diameter increment made it possible to get better results for the prognosis for a 4-year period, allowing the forest manager to better work with estimators either for diameter classes as well as for the forest to be managed. Using the data structure of trees (DBH ≥ 9.5 cm) of the National Forest of Sao Francisco de Paula, RS, Brazil, the diameter increment of species was stratified into four statistically different strata. Using data sampled between the years of 2000 to 2004, it was carried out prognosis for the year 2008, using the stratified and not stratified data. The estimates were compared with the observed values by the Kolmogorov-Smirnov (K-S) test, which attested goodness of fit only for the estimates from the stratified values. The prognosis estimated by means of stratified values also showed more accurate results, because the stratified variance was reduced nine times when compared with that obtained with the prognosis made without stratified data and to determine the Quotient of De Liocourt. Thus, the diameter growth stratification indicated to be effective to generate more accurate prognosis.

Brazil has the second largest forest area in the world, covering 516 million hectares and from this total 98.7% is made up of natural forests [

Methodologies consist of mathematical models to generate prognosis, as the ratio of moving diameters, which are dependent on information related to forest dynamics [

Whereas stratification has the goal to reduce the variance of a sample, resulting in subpopulations with greater homogeneity [

Although stratification of height is of great importance in phytosociological studies and in forest management and often used by managers [

The study was conducted in the National Forest of Sao Francisco de Paula, located between the coordinates 29˚24' and 29˚27' South latitude and 50˚22' and 50˚25' West longitude, in the mesoregion northeast of the State of Rio Grande do Sul, Microregion of Vacaria, in Brazil. The average altitude site is 900 meters above the sea level, ranging this value ±300 meters. The soils that characterize the region are classified as Haplumbrept, Argiudoll, Udorthent [

In accordance with the global classification of climatic types developed by Köppen, the climate of the region is of type Cfb, mesothermal and super humid, with mild summer and cold winter. The formation of frost is frequent, with snowfall in the colder months [

The predominant vegetation of the National Forest area is the Araucaria Forest or Mixed Tropical Forest, monitored by means of continuous forest inventory with the application of sampling with total replacement, which encompasses a total sample area of 10 permanent plots with an area of 1 ha (100 × 100 m) each, remeasured annually since the year 2000. All arboreal individuals (trees with diameter at breast height ≥9.5 cm) included in sample area were botanically identified and had their dendrometric information measured.

Taking the diameters collected in a specific period, their periodic increments were calculated and subsequently

the mean of the periodic increment

consistent, only those that met a minimum of 10 individuals in the sample were considered. After that it was defined the population of species E _{i} was measured, the (MPI) per individual of each species, such that it was characterized for each species a population of sample values P_{ij}, such that

The stratification of growth was established taking as criterion the quartiles formalized on the basis of the arithmetic mean of the growth of species _{h}, in that N_{h} is the number of species of each stratum and, consequently, _{jh}.

This procedure of stratification was decided as demonstrated by Cochran (1963), in that, if the full range of variation of MPI’s is expressed by (E_{L} − E_{0}), the separation of the strata (E_{h} − E_{h+}_{1}), being taken of constant size, will minimize the sum of squares due to stratification and, consequently, it will result in the minimum stratified variance of mean, i.e., maximum accuracy by application of stratification.

It was proposed four strata for growth evaluation: E_{1}, E_{2}, E_{3} and E_{4}, which is composed, respectively, of the smallest to the largest increments. The stratum E_{1} was composed by species with periodic mean increment _{2}, composed by the species with higher growth than E_{1} up to the limit of one standard deviation above that value_{3}, composed by species of growth higher than the E_{2}, up to the limit of one standard deviation above that value

fourth stratum E_{4} is understood to have the grater values than E_{3}, with mean

calculated the variance of the increments before stratification

Such parameters for the stratified population will be obtained as are expressed in (1) for the stratified mean and in (2) for the stratified variance.

The normality of the data set was checked by the Shapiro-Wilk test (W), and the homogeneity of variances by Hartley test (F_{max}). If null hypotheses H_{0} is rejected (normal distribution and homogenous variance), it will be applied to the data set the appropriate transformations using the method of Box and Cox, defined as:

where:

Twenty one values were selected for _{max}. As the set of observed data occurs in the interval (0, 1), taking

Due to stratification, an allocation of sample intensities became different per stratum and, consequently, with different degrees of freedom in each one of them; that is why it has been decided to use the Satterthwaite’s method to calculate the number of degrees of freedom [

where:

N_{h} = potential number of species (E_{h});

n_{h} = number of species allocated per stratum;

The value of

To carry out the statistical analysis between the strata means, it was applied the analysis of variance (ANOVA), whereas a completely randomized design with four treatments (strata) and different number of repli- cates, followed by application of Tukey’s test, when identified significant statistical differences between strata means.

The methodology used to accomplish the prognoses was the ratio of moving diameters, used as a theory of stand projection tables applied to native forests [

With the purpose to verify the effect of stratification on the results of the prognosis, it was first predicted the forest structure, employing the stratified data and, subsequently, it was held the prognosis with the raw data, without stratification. Both prognoses were made for the year 2008. The forecasted values were compared with the observed ones by application of the Kolmogorov-Smirnov test (K-S) at 95% probability, whose null hypothesis (H_{0}) consists in goodness of fit between projected and observed data.

Complementary to the goodness of fit test, due to its importance in structural evaluations and in the effectiveness of the forest management practices, it was evaluated the behavior of the De Liocourt Quotient (q) between the estimated and observed values.

As population of species was defined E_{i} was calculated

such that these mean values became associated with all species of the population E

The stratification of diameter growth was established taking as criterion the quartiles based on the average growth of the species _{1} = 8; N_{2} = 29; N_{3} = 18 and N_{4} = 11.

The periodic mean diameter increment ^{−1} found by Figueiredo Filho et al. in its bibliographical searches related to studies conducted in the Araucaria Forest [

. Synthesis of the methodology used for the prognosis in the forest before stratification

Center of class (cm) | Number of trees | Movement ratio^{1} | Remaining (%)^{2} | Mortality^{3} | Recruitment^{4} | Prognosis |
---|---|---|---|---|---|---|

15 | 2863 | 9.5733 | 90.4266 | 311 | 419 | 2696.9 |

25 | 1251 | 11.9912 | 88.0087 | 70 | 1031.1 | |

35 | 589 | 13.0108 | 86.9891 | 26 | 486.5 | |

45 | 333 | 14.4637 | 85.5362 | 16 | 269.0 | |

55 | 233 | 18.9068 | 81.0932 | 5 | 184.2 | |

65 | 117 | 18.0459 | 81.9540 | 1 | 95.2 | |

75 | 61 | 20.5915 | 79.4084 | 1 | 47.8 | |

85 | 21 | 16.8126 | 83.1873 | 0 | 17.9 | |

95 | 9 | 18.6129 | 81.3870 | 0 | 7.6 | |

105 | 8 | 22.5408 | 77.4591 | 0 | 7.1 |

^{1}indicates the ratio of the average growth by the width of the diameter class; ^{2}trees that remained in the same class; ^{3,4}number of trees that died and were recruited, respectively, during the observed period.

vironment [

Taking as basis for stratifying the average growth rate _{MPI} = 0.071 cm×year^{−1}), the limits of the strata were defined as: E_{1} for the values of_{2} for values of_{3} for the values of _{4} for the values of

The stratum of greater representativeness was the E_{2}, corresponding to 44% of the species evaluated, followed by E_{3} (27% of species), E_{4} (17% of species) and E_{1} (12% of species). The growth mean values of the respective strata were:

The variance for the population without stratification resulted in

Each stratified data set followed the normal distribution, evaluated by the Shapiro-Wilk test (W) at 95% probability. The effective number of degrees of freedom calculated by Satterthwaite’s method was 41 (n_{e} = 40.4). The variances were not homogeneous (heteroscedasticity), with rejection of the null hypothesis H_{0} by Hartley’s test (F_{calc} = 5.49 > F_{tab}_{;0.01} = 2.97). The data set was then subjected to Box-Cox’s transformation, with a value of _{calc}, ensured the homogeneity of variances (F_{calc} = 2.58 <

. Stratification of diameter growth by species in Araucaria Forest (E)

E_{1} N_{1} = 8 W_{1} = 0.12 |
---|

Acca sellowiana (O. Berg) Burret (0.007); Symplocos uniflora (Pohl) Benth. (0.014); Myrrhinium atropurpureum Schott (0.018); Myrcianthes gigantea (Cambess.) O. Berg (0.027); Dasyphyllum spinescens (Less.) Cabrera (0.035); Campomanesia rhombea O. Berg (0.037); Myrcia oligantha O. Berg (0.050); Eugenia uruguayensis Cambess. (0.052) |

Xylosma pseudosalzmannii Sleumer (0.058); Eugenia involucrata DC. (0.063); Pilocarpus pennatifolius Lem. (0.064); Lonchocarpus campestris Mart. ex Benth. (0.065); Myrceugenia miersiana (Gardner) D. Legrand & Kause (0.065); Myrceugenia cucullata D. Legrand (0.067); Roupala Montana Aubl. (0.068) Sebastiania brasiliensis Spreng. (0.070); Xylosma tweedianum (Clos) Eichler) (0.070); Machaerium paraguariense Hassl. (0.074); Calyptranthes concinna DC. (0.075); Eugenia psidiiflora O. Berg (0.075) Siphoneugena reitzii D. Legrand (0.078); Scutia buxifolia Reissek (0.078); Annona rugulosa (Schltdl.) H. Rainer (0.079); Myrciaria floribunda (H. West ex Willd.) O. Berg (0.081); Myrcianthes pungens (O. Berg) D. Legrand (0,083); Campomanesia xanthocarpa O. Berg (0.087); Citronella gongonha (Mart.) R.A. Howard (0.097); Sapium glandulosum (L.) Morong (0.098); Podocarpus lambertii Klotzsch ex Endl. (0.098); Lithraea brasiliensis Marchand (0.102); Sebastiania commersoniana (Baill.) L.B. Sm. & Downs (0.111); Myrsine umbellata Mart. (0.112); Luehea divaricata Mart. & Zucc. (0.114); Maytenus evonymoides Reissek (0.118); Myrsine coriacea (Sw.) R. Br. ex Roem. & Schult (0.121); Ilex dumosa Reissek (0.121); Eugenia subterminalis DC. (0.123) |

Blepharocalyx salicifolius (Kunth) O. Berg (0.131); Inga vera Kunth (0.133); Casearia decandra Jacq. (0.136); Casearia obliqua Spreng. (0.143) Dasyphyllum tomentosum (Spreng.) Cabrera (0.147); Myrsine sp. (0.150); Araucaria angustifolia (Bertol.) Kuntz (0.152); Cryptocarya aschersoniana Mez (0.160); Ilex microdonta Reissek (0.162); Ocotea porosa (Nees & Mart.) Barroso (0.163); Cryptocarya moschata Nees & C. Mart. (0.167); Zanthoxylum rhoifolium Lam. (0.167); Solanum sanctae-catharinae Dunal (0.168); Dicksonia sellowiana Hook. (0.175); Ocotea pulchella Mart. (0.182); Ilex brevicuspis Reissek (0.184); Zanthoxylum petiolare A. St.-Hil. & Tul. (0.196); Allophylus edulis (A. St.-Hil., Cambess. & A. Juss.) Radlk. (0.196) |

Cupania vernalis Cambess. (0.197); Prunus myrtifolia (L.) Urb. (0.207); Ilex paraguariensis A. St.-Hil (0.209); Laplacea acutifolia (Wawra) Kobuski (0.215); Matayba elaeagnoides Radlk. (0.225); Lonchocarpus nitidus (Vogel) Benth. (0.231); Lamanonia ternata Vell. (0.233); Ocotea puberula (Rich.) Nees (0.256); Vernonanthura discolor (Spreng.) H. Rob. (0.265); Nectandra megapotamica (Spreng.) Mez (0.289); Cinnamomum glaziovii (Mez) Kosterm. (0.322) |

F_{tab}_{;0.01} = 2.97) and remained normally distributed. Consequently, detected heteroscedasticity of the variances within strata in which the total population of

By means of analysis of variance, it can be observed that there are statistical differences between the strata (

The Tukey’s test indicated statistical differences between means of the strata (

Whereas the models of production meets a set of criteria that assist in decision making for the forest planning [

Meyer et al. in their studies that support the sustainable management, about the diameter distribution in hete-

Behavior of the value F_{max} as a function of the values of λ_{1} tested

. Analysis of variance for stratification in Araucaria Forest

FV | GL | SQ | QM | F_{calc} | F_{crit }_{99%} |
---|---|---|---|---|---|

Between strata | 3 | 0.123062 | 0.041020 | 180.22^{**} | 2.75 |

Within the strata | 62 | 0.014112 | 0.000227 | ||

Total | 65 | 0.137173 |

^{**}Significant at 1% level of probability.

. Tukey’s test at 95% probability between the strata means

Strata | Means | |||
---|---|---|---|---|

Transformed | Retransformed | Observed | ||

E_{1} | −0.25082 | 0.02937 | 0.03000 | a^{*} |

E_{2} | −0.19736 | 0.08569 | 0.08672 | b |

E_{3} | −0.14600 | 0.16089 | 0.16178 | c |

E_{4} | −0.10932 | 0.23798 | 0.24082 | d |

^{*}Means followed by same letter do not differ from each other by Tukey’s test at 95% probability.

Projected frequency distribution per stratum and for the population

rogeneous all age forests, refer to a pattern of decreasing exponential distribution to the extent to which diameter classes advance [

The K-S test (D_{calc} = 0.00949 < D_{tab}_{;0.95} = 0.01855) attested goodness of fit of projected values from stratified data, however the projected values from no stratified data did not present goodness of fit to the observed distribution (D_{calc} = 0.09826 > D_{tab}_{;0.95} = 0.01855). Therefore, results obtained with greater homogeneity between diameter growth values within strata tend to provide better estimates because the variations of growth between species are minimized with the group formation, allowing a refinement of estimates when using the results of the stratified parameters. This confirms our formulated hypotheses. The distributions of projected and observed frequencies are presented in

De Liocourt considers that in a balanced forest, the diametric distribution in successive classes (k) is derived from a constant geometric series called “Quotient of De Liocourt” (q), essential parameter for the forest manage- ment proposal [

The prognosis of the diametric structure, in these circumstances, in addition to estimate with accuracy the forest density, must maintain values equally accurate for obtaining the quotient q, indicating an approximate results for quotients of distribution of frequencies for the forest structure, which can be observed in

In both projected distributions and in the observed one, the q quotients, minimum and maximum, remained in the same classes, k_{4}/k_{5} and k_{7}/k_{8} respectively. The lowest accuracy between the projected and observed ratios was found in the estimation without stratification, for the lower and upper diameter classes, reaffirming what was observed in the distribution of frequencies. Generally speaking, disregarding the lower and the upper diameter classes without stratification, there is similarity between the estimated distributions and the observed one, essentially when performed with stratification. Consequently, the prognosis of diametric distribution for the year 2008 resulted more accurate when calculated from stratified data, evaluated by the goodness of fit attested at 95% probability by the application of the Kolmogorov-Smirnoff test in relation to data observed in 2008; the same condition occurred in the evaluation of the Quotient of Liocourt (

Other methodologies can be developed to fit the models of prognosis, providing greater accuracy in their results. In heterogeneous forests, for example, it will be appropriate to separate groups formed by species or botanical families. The application of probability density functions can also improve the prognosis, favoring a distribution with greater similarity between estimated and observed values of the forest.

The auto ecological characteristics of each species that composes the plant community favor decisively the formation of groups with different diameter growth and the stratification of diameter growth reduced the variance nine times, enabling the formation of groups statistically different of greater homogeneity.

The stratification of diameter growth approached the predicted results to those observed in the sample population, allowing the forest manager to work with the best estimators in preparation of management proposals.

The methodology of ratio of change, which is based on diameter’s growth and migration of them between

Distribution of De Liocourt quotients for observed and predicted values

. Prognosis of the number of individuals per diameter class generated from the values with and without stratification, and observed values of the diametric distribution of Araucaria Forest, for the year 2008

Center class (cm) | With stratification | Without stratification | N observed |
---|---|---|---|

14.5 | 2657.2 | 2696.9 | 2619 |

24.5 | 1248.5 | 1031.1 | 1263 |

34.5 | 592.9 | 486.5 | 643 |

44.5 | 349.8 | 269.0 | 351 |

54.5 | 234.9 | 184.2 | 245 |

64.5 | 129.5 | 95.2 | 140 |

74.5 | 61.2 | 47.8 | 64 |

84.5 | 28.0 | 17.9 | 26 |

94.5 | 13.0 | 7.6 | 13 |

≥104.5 | 8.0 | 7.1 | 7 |

∑ | 5323.1 | 4843.2 | 5371 |

. Values for the quotient of De Liocourt from the prognosis of diametric distribution and observed values of Araucaria Forest, for the year 2008

Classes | Quotient of Liocourt (q) | ||
---|---|---|---|

With stratification | Without stratification | Population quotients^{*} | |

k_{1}/k_{2} | 2.13 | 2.62 | 2.07 |

k_{2}/k_{3} | 2.11 | 2.12 | 1.96 |

k_{3}/k_{4} | 1.69 | 1.81 | 1.83 |

k_{4}/k_{5} | 1.49 | 1.46 | 1.43 |

k_{5}/k_{6} | 1.81 | 1.93 | 1.75 |

k_{6}/k_{7} | 2.12 | 1.99 | 2.19 |

k_{7}/k_{8} | 2.19 | 2.67 | 2.46 |

k_{8}/k_{9} | 2.15 | 2.36 | 2.00 |

k_{9}/k_{10} | 1.63 | 1.07 | 1.86 |

q | 1.92 | 2.00 | 1.95 |

^{*}Values obtained with all population data.

diameter classes showed to be very efficient to make prognosis in native forests.

The prognosis of diametric distribution for the year 2008 resulted more accurately when calculated from stratified data, evaluated by the goodness of fit attested at 95% probability by the application of the Kolmogorov-Smirnoff test in relation to data observed in 2008.

The evaluation of the coefficient of De Liocourt behaved better in stratification using diameter growth by class- es when compared to the results obtained without stratification.