Introduction

The accuracy in calculating the volume of standing timber is essential for sustainable forestry, for which there must be tools to obtain such information from a tree or mass quickly and reliably, by measuring variables easy to get as the normal diameter (dn) or the total height (h) (^{Barrio et al., 2004}).

The evolution of forest stands in their mensuration characteristics resulting from silvicultural management modifies in proportionally the shape of the trees, the wood quality and the distribution of its products (^{Uranga et al., 2015}), which are inherently related to site conditions, density and status of each individual (^{Harold and Hocker, 1984}). Therefore, it is important to generate new volume tables and form factors (FF) to avoid errors in estimating timber stocks, disrupting the annual cutting programs and, in general, the planning and execution of forest management (^{Carrillo et al., 2004}).

A morphic coefficient or form factor is defined as the quotient between the actual volume of the trunk and the volume of a model that is taken as a reference and is highlighted with the letter f; meanwhile a quotient so is the expression between two diameters measured at different heights of the trunk, one reference diameter and another at a higher height and are denoted with the letter k (^{Diéguez et al., 2003}; ^{Prodan et al., 1997}).

In the Comunidad Indígena de Nuevo San Juan Parangaricutiro, Michoacán State, a silvicultural management of the forest has been applied since 1983, which, along with other factors, have changed the characteristics of tree stands over time, which in turn has impacted the shape of the trees and the distribution of its products as in situations observed by ^{Van Laar and Akça (2007)}, ^{Picard et al. (2012)} and ^{Uranga et al. (2015)}. As quoted before and, because authors like ^{Santiago et al. (2013)} indicate changes in the production and yield of forest stands according to age, it is considered important to carry out new studies to estimate the timber volume with updated tools. Under the need for this information, the objective of this study was to evaluate the accuracy of the coefficients and quotients as a means to make the calculations of the volume of the trunk of standing trees of Pinus montezumae Lamb., in forests of the population referred.

Materials and Methods

Sampling and data collection

Choice and sample size. Healthy trees, with straight and clear bole were chosen, which were not isolated or on the edge of the stand. A preliminary sampling of the mass was performed to determine the range of the existing diametric categories. In literature it is suggested between 23 and 229 trees for local and regional tables or for an equation of volume (^{Da Cunha and Guimarães, 2009}; ^{Honorato, 2011}). Between 8 and 16 trees in each diameter category were selected, which were referred to as “type trees”; a total of 245 trees were used, from which 161 were taken to generate an equation of volume and 84 samples to calculate the morphic coefficients and quotients.

Measurement of variables. The diameters of the base, stump height and 1.3 m in the shaft were measured with a caliper (Haglöf Mantax Blue 80 cm), the total height and diameters at different heights with a Bitterlich relascope^TM (Pat. No. 172305). The volumes corresponding to different sections of the trunk were calculated with the Smalian formula and the tip with that of the cone (Table 1). The stem volume (VFU, from its acronym in Spanish), by adding the volume of all sections of the tree.

Table 1 Expressions for calculating the volume of the trunk sections. 

Where:

ab _t = Basal area of the stump (m²)

ab ₀ = Basal area of the largest section (m2)

ab ₁ = Basal area of the smallest section (m2)

ab _n = Area of the basis of the tip (m2)

LS = Length of the log or section (m)

LP = Length of the tip (m)

Calculation of the morphic coeficients (f) and shape quotients (k)

In order to make the identification of the trees easier, the central axis was named “trunk” and “bole” the length of the axis up to the thin tip of 10 cm (^{Diéguez et al., 2003}). The calculation of the morphic coeficients and shape quotients was made with the volume of each tree with the equations shown in Table 2.

Table 2 Equations and notation of the coefficients and quotients evaluated to estimate the volume of standing Pinus montezumae Lamb. trees. 

Statistical analysis

Detecting outliers. With the Statgraphics Centurion package and the “Improve-Regression Analysis Multiple Regression -Several factors” (^{StatPoint, 2005}), outliers that negatively influenced the adjustment equations were detected. The comments were removed with Studentized >2 residues in absolute value, with the following expression:

If RStudent ≤ 2, assess if it is greater than 2 and in such a case, eliminate observations.

Correlation and regression analysis. Mathematical models were fitted to predict the volume of standing trees, with the SAS v. 9 statistical package (^{SAS, 2009}). In the adjustment, the following models, Schumacher-Hall, Thornber, Spurr with Arithmetic Combined Variable, Spurr with Logarithmic Combined Variable and Korsun (Table 3) were used; these statistics were compared: mean error (CME) square, fitted coefficient of determination (fitted R2) and the significance of the model by calculated F and the parameters obtained. In addition, the dispersion of the residuals, the autocorrelation of errors by the Durbin-Watson and data normality with Shapiro- Wilk test were graphically analyzed.

Table 3 Mathematical models used to estimate the volume of Pinus montezumae Lamb. 

The volumes that were calculated with the best fit equation were compared with the volumes obtained with each of the morphic coefficients and / or quotients of shape, and evaluated by the deviations expressed in terms of the Aggregate Difference (DA, for its acronym in Spanish) as a percentage. The decision rule was: the volume that came from the form coefficients or quotients with DA close to 1 % can be used reliably; also the range and standard deviation of the observations was considered. Based on the works of ^{Romahn et al. (1994)} and ^{Fonseca et al. (2009)}, the expression to calculate the Aggregate Deviation is:

Where:

DA = Aggregate Difference in percentage

ΣVc = Sum of the calculated or estimated volumes

ΣVr = Sum of the real volumes

Results and Discussion

Sample size. For the volume equation, the final size of the sample after removing outliers, trees was 157 and for calculating the morphic coefficients and quotients 84 were used. Some authors take different number of samples for building volume models, including ^{García et al. (2012)} who used 207 P. michoacana and 220 P. michoacana var. cornuta Martínez; ^{Lores et al. (2010)} worked with 23 samples of Calophyllum brasilense Cambess and ^{Montes de Oca et al. (2009)} with 124 trees of Pinus duranguensis Martínez. The sample used to generate both, the volume equation as well as the form factors, complies to support reliable results.

Regression analysis and correlation. The Schumacher-Hall and Thornber models had low values in the mean squared error (MSE) (CME, for its acronym in Spanish). The probability value is significant (Pr ˂0001) for all models as well as the explanation of the sample by them that exceeds 99 %. The results of the regression analysis and statistical indicators of goodness of fit are presented in Table 4.

Table 4 Summary of statistical results of each tested model. 

GL = Degrees of freedom; CME = Mean squared error.

Graphical distribution of residuals. The analysis of distribution of the residuals of the best models to determine the nonexistence of collinearity was performed, that the errors are uncorrelated, that they are normally distributed and that error has a uniform variance (^{Alder, 1980}; ^{Velasco et al., 2007}; ^{Da Cunha and Guimarães, 2009}); in addition to visually examine how relationships work and to corroborate their homocedasticity or heteroscedasticity (^{Prodan et al., 1997}); the analysis revealed no trends (Figure 1).

Figure 1 Distribution of residuals. 

The Schumacher-Hall and Thornber Models revealed statistical and graphical results of similar residuals; the first was preferred by the parsimony of it, easier calculations and for being the most used in coniferous forests.

Validation of the Schumacher-Hall model. In Table 5 are the results of the Durbin-Watson and Shapiro-Wilk statistics and the significance of the β₀, β₁ and β₂ parameters from the statistical fit.

Table 5 Validation statistics of the Schumacher-Hall model. 

The Durbin-Watson (DW) statistic points out that there is no autocorrelation of errors, because it has a value of 1.96 in which residues are independent; the Shapiro-Wilk statistic with 0.995111 indicates that the assumption of normality of errors is not violated because the probability of rejection is less tan 0.05 and is significant (^{Martínez et al ., 2006}; ^{Velasco et al ., 2007}; ^{Da Cunha and Guimarã es, 2009}) so it is assumed that the sample comes from a population that follows a normal distribution; Table 2 shows that the coefficients β₀ , β₁ and β₂ are significant (p> F = <0.0001). The equation generated with the model of Schumacher-Hall was:

V=0.0000584616 D^1.96205 H^0.93483

Assessment of the morphic coefficients and quotients. With the volume of the 84 trees set aside for in particular to make the calculation of the coefficients and quotients involved, an average resulted, which is ordered in Table 6.

Table 6 Average values of the morphic coefficients and quotients. 

The value of the morphic coefficient of the trunk varies between 0 and 1 as the volume of any dendrometric type alone, or a combination of several, is always less than or equal to the volume of the cylinder (^{Diéguez et al., 2003}). The value obtained in this study was 0.525 for Pinus montezumae, however it is higher than that of ^{Uranga et al. (2015)} whose value varied between 0.44 and 0.50 for P. patula Schiede ex Schltdl. & Cham. in three regions of Mexico.

The morphic coefficient of the bole at the total height is the most used when trying to estimate the stem volume; however, is that its value has no upper limit set, which is rather inconvenient; that is, it can be > 1, as the volumes of two bodies of different heights are compared, but in any case, it is always <1 for classic dendrometric types (^{Dieguez et al., 2003}); for P. montezumae the average value was 0.557.

For the morphic coefficient of the bole at the height of the bole, the advantage is that the volumes of two bodies of equal height, are compared, whereby the upper limit value = 1 (^{Diéguez et al., 2003}); the average value in this study was 0.601.

The morphic coefficient referred to the basal section is rarely used due to irregularities at the trunk base. Those corresponding to the fixed sections have the disadvantage that for trees of equal shape, that is, trees which decrease in the same proportion along the longitudinal axis, can give different values and vice versa, trees with different shape can have the same morphic coefficient value (^{Diéguez et al., 2003}). The calculated average for P. montezumae is 0.460.

False morphic factors are not fully comparable as they do not allow a direct representation of the geometrical shape of the bole due to its reference to d, and contain a distorting element (^{Prodan et al., 1997}).

Two trees with the same shape have equal referred morphic coefficient to a relative (not fixed) section; therefore, they are more descriptive of their form than artificial coefficients. The values for Hohendahl’s coefficients for classic dendrometric types were: cylinder (1), paraboloid (0.556), cone (0.411) and neiloide (0.343) (^{Diéguez et al., 2003}). Hohendahl’s method outstands because it leads to a shape factor naturally by measuring two diameters of the bole always in relative terms, or allowing be comparable in any of the dimensional conditions of the trunk. The result of the average Hohendahl’s morphic coefficient to the trunk was 0.561, with the bole to total height, 0.587 and of the bole to the height of it, 0.639; when comparing these three values with those suggested by this author for the dendrometric types mentioned above, the shape of the trunk of P. montezumae resembles a paraboloid.

When applying Pressler’s formula for the dendrometric types, the generated data were the following: paraboloid, 0.500; cone, 0.333 and neiloide, 0.247 (^{Diéguez et al., 2003}); the average obtained in this work is 0.511, a value that confirms the type of paraboloid trunk of P. montezumae; in the first two, exact volumes are obtained and in the case of neiloide, it reveals slightly lower values than the real ones (98.6 %); however, doubts about their reliability and applicability arise when passing from the concept of geometric solids to real logs tree (^{López, 2005}).

The handicap of Schiffel and Pollanchütz quotients is that they do not have an upper limit value, being able to exceed the cipher = 1, because if the tree height is less than 2.06 m or 4.33 m respectively, the part where the diameter of the numerator is measured is below the normal height, and therefore, the ratio can be > 1. The disadvantage compared to the normal Schiffel’s form factor is that it does not provide information on the volume under the normal diameter (^{Diéguez et al., 2003}); the average form quotient of Schiffel obtained for P. montezumae was 0.718. The values from Pollanchütz morphic quotients for the dendrometric types were: paraboloid, 0.882: cone, 0.788, and neiloide, 0.686; in this study the average was 0.831, confirming the paraboloid shape of the trees (^{Prodan et al., 1997}).

Johnson’s shape quotient represents more accurately the trunk form than the normal morphic quotient of Schiffel, which has a maximum value of one, which is reached when the tree has a height of 1.30 m and in most species its value varies between 0.6 and 0.8. Johnson‘s morphic quotient for the dendrometric types are: paraboloid (0.707), neiloide (0.354), cone (0.500) (^{Diéguez et al., 2003}), the average obtained was 0.736 for P. montezumae that is within domain range of this ratio. The average form quotient of Gieruszinski is 0.581 and found to be the lowest value of all the evaluated quotients.

With P. montezumae of the region, the sensitivity of the quotients is lost due to the paraboloidism of logs, which is more pronounced in young trees. The quotients so used were those called “false”, which are referred to a fixed section; the values here obtained were varied as they are not at percentage of their total or timber height, thus the ratio diameter-height is lost.

Aggregate Deviation and Standard Deviation of morphic coefficients

Deviation aggregate (DA) and Standard Deviation. The criterion of the DA states that between two estimates the difference should not exceed ± 1 % to show a reliable relationship (^{Romahn et al., 1994}; ^{Prodan, 1997}; ^{Fonseca et al., 2009}); in the present case, it refers to the estimated volume values and those obtained with the morphic coefficients and quotients. Standard deviation is the average deviations of the observations in regard to their mean expressed in the same units of measurement (^{Caballero, 1972}) so that the form coefficient or quotient that records the lowest value is more closely aligned to the central value and therefore it is more efficient (tables 7 and 8).

Table 7 Aggregated Deviation (%) and Standard Deviation of the morphic coefficients. 

Table 8 Aggregated Deviation (%) and Standard Deviation of the morphic quotients.  

With the morphic coefficient of the trunk (52 %) and the morphic coefficient of the bole to the total height (-0.85) se the percent premise of ± 1 % was achieved, the obtained coefficient with the Pressler formula reached a value rather close to one (1.18 %) and the rest of the coefficients were between 6.98 % (morphic coefficient for the bole at the height of the bole) and 15.18 % (Hohendahl’s morphic coefficient for the bole at the height of the bole).

The coefficients that had the smallest standard deviation were that of the bole at total height (2.2013) and that of Pressler (2.2126); the highest, the referred to the basal section (3.0903) and Hohendahl’s morphic coefficient for the bole at the height of the bole (2.5631).

Aggregate Deviation and Standard Deviation of morphic quotients

The values in DA obtained by the shape quotients were significantly higher than those from the form coefficients as their numbers varied from 28.22 % (Schiffel) to 43.42 % (Zimmerle).

The standard deviation of the morphic quotients that showed smaller percentage were Johnson’s, Schiffel’s and Pollanschütz’s; they highest were those of Zimmerle and Gieruszinski. Even though the morphic coefficients and quotients that use the bole in thin tip are similar to those suggested by their authors, it is likely that its accuracy increases if the diameter in thin tip also makes it from 10 to 20 cm or more, which may vary with the dendrometric type that better assimilates the trunk of a particular species.

Of all the coefficients and quotients only three achieved the precision required to make reliable estimates of the volume of standing trees: the morphic coefficient of the trunk and the morphic coefficient of the bole to total height as well as the coefficient obtained with the adjustment in the Pressler formula.

Comparison of the calculated volumes with the equation of volume and form coefficients

In Figure 2, the graphs illustrating the relationship between the volumes obtained with Schumacher- Hall’s model are presented as well as those resulting from the best shape coefficients.

Figure 2 Comparison between the volumes of the form coefficients and quotients and the volume equation. A. Morphic of the trunk, B. Morphic coefficient of the total height, C. Pressler’s morphic coefficient, D. Hohendahl’s morphic coefficient for the trunk and E. Hohendahl’s morphic coefficient for the bole at the total height. 

In the first three cases there is broad agreement between the estimated values from the volume equation and those obtained with the indicated coefficients, demonstrating the feasibility of using them reliably to estimate the volume of P. montezumae standing trees growing in the forest of the Comunidad Indígena de Nuevo San Juan Parangaricutiro, Michoacán. Values from Hohendahl’s morphic coefficient for the trunk and the bole overestimate volumes from 20 cm and 50 cm diameter classes, respectively.

Conclusions

Morphic coefficients and quotients are an essential tool for a quick and accurate estimate of the volume of standing trees and the use of one of them depends on the shape of the trunks of the species under evaluation; quotients provide a better estimation of the conic and neiloide shapes and coefficients associate in a closer way to the paraboloid. In addition, they are an alternative for building commercial volume tables for individuals of this species.

False morphic coefficients allow a more accurate estimation of volume from the paraboloidism of Pinus montezumae, while the real ones use the reference diameter in regard to the total height of the tree, which, in order to be more precise, require a sharper taper than that of the species.

The average values from Hohendahl’s morphic coefficient for the trunk and the bole at total height, are within the ranges suggested by this author for the paraboloid dendrometric type (0.566); however, when comparing them with the real volumes, an overstimation is found.

The morphic coefficients of the bole, of the bole at total height and Pressler’s morphic coefficient were those with minor deviations between estimated and calculated volumes so its use is reliable in the volumetric estimation of tree trunks of Pinus montezumae.

From the results obtained it can be concluded that it is possible to predict the shape and volume of Pinus montezumae trunk by morphic coefficients and quotients resulting in a paraboloid shape.

Conflict of interests

The authors declare no conflict of interests.

Contribution by author

Guadalupe Geraldine García Espinoza: formulation of the research study, field data collection, planning, writing and structuring of the manuscript; José Jesús García Magaña: planning, design and supervision of field work, writing and structuring of the document; Johnathan Hernández Ramos: data taking and analysis and writing of the document; Hipólito Jesús Muñoz Flores: training in handling Bitterlich’s Telerelascope and review of the manuscript; Xavier García Cuevas: statistical advice and review of the manuscript; Adrián Hernández Ramos: field data analysis and review of the manuscript.