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Introduction
Timber volume estimation allows us to infer the growth and development of forest stands, through the application of silvicultural treatments, with the intention of improving the quality of each of the individuals that comprise them (Diéguez-Aranda et al., 2009). In this sense, forest inventories are tools that allow for the evaluation of the resources available in forests, since they provide an overview of their current situation and enable decision-making on management, conservation, restoration or use of the trees and the commercial distribution of the products (Alvis-Gordo, 2009; Corral-Rivas & Návar-Cháidez, 2009; Del Rio, Montes, Cañellas, & Montero, 2003; El-Juhany, Aref, & El-Wakeel, 2002).
Timber stocks can be quantified by both direct and indirect methods. The xylometer technique is based solely on the direct method (destructive sampling) (Aldana-Pereira, 2008; Cruz-Cobos, Mendía-Santana, Jiménez-Flores, Nájera-Luna, & Cruz-García, 2016), while the Pressler (or guidepoint) and standing tree volume techniques, dendrometric bodies, rigorous volume estimation, the graphical method, conventional equations and artificial neural networks can be calculated using both methods.
Several equations estimated by the direct and indirect methods allow the quantification of timber stocks in natural forests or in plantations of mainly commercial interest, through forest inventories. Mathematical models are tools that allow representing a phenomenon through an equation, facilitating the interpretation of the prediction about the future behavior of an event, for making decisions about resource management. The application of these models guarantees optimal results in timber harvesting work (Diéguez-Aranda et al., 2009). In this context, equations are essential for modeling the variety of relationships between response and estimated variables. Nonlinear (logistic and exponential) equations are ideal to describe biological and physical systems of species (Urbano, Machado, Figueiredo-Filho, Sanquetta, & Zea-Camaño, 2018).
In tropical ecosystems there is a wide diversity of species with morphological structures that contribute to growth and development conditions (Guariguata et al., 2009; Martínez-Tobón, Aunta-Duarte, & Valero-Fandiño, 2013); for example, the tree stem presents difficulties in terms of quantifying forest volume and, consequently, for the distribution of products (Cancino-Cancino, 2012; De Azevêdo, Paes, Calegari, & Do Nascimento, 2014). Given the scarcity of information on volume modeling in tropical species and its importance in forestry, especially in the commercial sector, the objective of this work was to investigate volume estimation techniques in the tropics.
Materials and methods
The paper was developed with the non-probabilistic snowball technique, in which the bibliography reported in an investigation suggested the search for other similar works to enrich the subject (Baltar & Gorjup, 2012; Vásquez-Bautista, Zamudio-Sánchez, Alvarado-Segura, & Romo-Lozano, 2016). For this, books, scientific journals in the EBSCOhost, Scopus, Google Scholar and Elsevier search engines, and libraries of various educational institutions were explored.
The search focused on locating volume equation systems for forest species in tropical zones; in addition, the digital library of the Forest Planning System (SiPlaFor) platform for temperate forests was viewed. Some keywords were used such as: trees, volume estimation, equations, tropical species, estimation methods, accuracy and volume. Subsequently, volume and yield estimation techniques were compared to analyze their application and predictive capabilities.
Results and discussion
Direct and indirect methods
The equations estimated by the direct and indirect methods allow the quantification of timber stocks in natural forests or in plantations of mainly commercial interest, through forest inventories. Based on careful analysis, it can be deduced that the xylometer technique is the only one that can be considered as a direct method (destructive sampling) for volume estimation, since the other techniques can use both methods; the use of the method is at the discretion of the researcher when planning fieldwork.
Each method has its advantages and disadvantages; for example, the indirect method is very practical since it does not imply felling the tree and the variables to be used can be obtained with available tools, such as the Bitterlich tele-relaskop, TruPulse hypsometer or Criterion RD1000 and Barr-Stroud dendrometers (Costas, Veran, Lorán, López, Fosco, & González, 2006; Gómez-González, Domínguez-Domínguez, Martínez-Zurimendi, & Ramírez-Valverde, 2018; Valencia-Manzo et al., 2017; Williams, Cormier, Briggs, & Martínez, 1999). In the direct method, tree measurement information can be obtained through destructive sampling and by means of mountaineering or forest bicycle equipment, telescopic, poles, grade rods, and the use of a diameter tape or tree caliper, with which the diameters are measured at different heights (Diéguez-Aranda et al., 2003). The same equipment can be used in the indirect method. Destructive sampling consists of felling previously selected individuals; because the measurement process can be carried out very easily, the quality of the information obtained is slightly better than that generated with the indirect method and with the equipment (Melo & Lizarazo, 2017).
Xylometer technique
Although not frequently used, this technique, also known as Archimedes' displacement, is the most accurate for estimating the volume of individual trees through the displacement of water caused by submerging each of the pieces in a previously calibrated container (xylometer) (Food and Agriculture Organization of the United Nations [FAO], 1980). This method is based on the principle that a body submerged in a static fluid tends to receive a vertical (upward) thrust equal to the volume of fluid displaced by the object (Romahn de la Vega & Ramírez-Maldonado, 2010). The volume is calculated with the following formula:
Martin (1984) concluded that this technique obtained excellent results compared to rigorous volume estimation techniques when experimenting with various broadleaf and shrub species; likewise, Biging (1988) supported this information. On the other hand, Figueiredo-Filho and Budant-Schaaf (1999) indicated that the xylometer technique is more accurate than conventional methods, and even more so when fitted to taper-volume equations; however, despite the high reliability, the use of the xylometer is limited since its application lies in areas with shrub or scrub species that do not have a defined stem (Aldana-Pereira, 2008), while in large logs it is difficult to use (Cancino-Cancino, 2012).
Pressler and standing tree volume formulas
Most of the time, the Pressler (or guidepoint) and standing tree volume formulas have been applied to unfelled trees, as they provide quality information. Volume tables generated with Pressler's formulas provide excellent results for cone-shaped or paraboloid solid bodies and allow estimating the stem without having to divide it (Gómez-González et al., 2018); however, these have been displaced by mathematical models (Da Cunha & Guimarães-Finger, 2009), due to the difficulty in measuring the height at which half the diameter at breast height is found. The standing tree volume formula consists of measuring the diameter at breast height (1.30 m from ground level) of a representative sample of trees, with the aid of a tree caliper or diameter tape; subsequently, a morphic coefficient is used to estimate the volume, although this factor often does not correspond to the species (Hazard & Berger, 1972).
The Pressler (or guidepoint) and standing tree volume equations are expressed, respectively, as follows:
where,
V |
volume (m3) |
g |
basal area (m2) |
hp |
guidepoint height (m) |
d |
diameter at breast height (cm) at 1.30 m above ground level |
mh |
merchantable height (m) |
h |
total height (m) |
f |
form factor |
e |
model error. |
The form or morphic coefficient consists of dividing the merchantable or actual volume of the stem by the volume of the conventional geometric body of the same base as the diameter at breast height section of the tree and its same height (Ugalde, 1981). The value of the coefficients depends on the forms of the stem, which can be neiloid, cylindrical, parabolic or conical, with the cylindrical form factor being the most commonly used (Cancino-Cancino, 2012). This coefficient is always less than 1 for the classic dendrometric types (Diéguez-Aranda et al., 2003).
The volume of standing trees can be estimated in plantations or natural forests using the morphic coefficient with excellent results, which makes it possible to prepare a standardized volume table for the species of interest (Michela, Kees, & Skoko, 2018). It is important to highlight the above, since in some works the 0.75 coefficient is still used for several pine species and some broadleaf or tropical ones (Rojas-Gutiérrez, 1977), which overestimates or underestimates the volume, because the coefficient varies with the dimensions of the stem and the species (Aguilar, Sequeira, & Peralta, 2017) and therefore does not always generate confidence (De Azevêdo et al., 2014).
Estimation by dendrometric bodies
Environmental interactions, as well as forestry activities, are responsible for the shape of a tree's trunk, mainly in most tropical species, where the base of the tree resembles a parabola known as a neiloid that changes to a cylinder, while the middle part takes a shape called a paraboloid and the tip resembles a cone (Cancino-Cancino, 2012). Given this, criteria are taken for field measurements to obtain reliable information, where stems bifurcated below the diameter at breast height (1.30 m) are considered to be two trees and averaged, while those bifurcated above 1.30 m are considered to be a single tree (Cruz-Cobos et al., 2016). In these cases, the tree sections are determined using solids of revolution. The equations for the dendrometric bodies are as follows:
Neiloid
Cylinder
Paraboloid
Cone
where,
V |
volume (m3) |
gi |
sectional area of one end of the section (m2) |
H |
height in one third (m) |
e |
model error. |
Stems that resemble the neiloid shape have a form factor of less than 0.38; cylindrical stems can take values equal to or greater than 0.75; paraboloid-shaped ones vary from 0.40 to 0.74; and those resembling a cone range from 0.27 to 0.39. Table 1 indicates some tropical species of commercial interest according to the geometric figures that the stem represents (Hernández-Ramos et al., 2018; Rojas-Gutiérrez, 1977; Moras & Vallejos, 2013; Tewari, Mariswamy, & Arunkumar, 2013).
Table 1 Tropical species with defined dendrometric bodies.
| Species | Form |
|---|---|
| Carapa guianensis Aubl. | Neiloid |
| Hymenaea palustris Ducke | Neiloid |
| Pterocarpus officinalis Jacq. | Neiloid |
| Clathrotropis brunnea Amshoff | Neiloid |
| Cordia alliodora (Ruiz & Pav.) Oken | Neiloid |
| Fraxinus chinensis Roxb. | Neiloid |
| Eucalyptus sp. | Neiloid |
| Prioria copaifera Griseb. | Cylinder |
| Brosimum utile (H.B.K.) Pittier | Cylinder |
| Humiriastrum procerum (Little) Cuatrec. | Cylinder |
| Huberodendron patinoi Cuatrec. | Cylinder |
| Tectona grandis L. f. | Cylinder |
| Swietenia macrophylla King | Cylinder |
| Cedrela odorata L. | Paraboloid |
| Protium neglectum Swart | Paraboloid |
| Campnosperma panamense Standl. | Paraboloid |
| Tapirira guianensis Aubl. | Paraboloid |
| Cariniana pyriformis Miers | Paraboloid |
| Jacaranda copaia (Aubl.) D. Don | Paraboloid |
| Guarea trichilioides L. | Paraboloid |
| Couma macrocarpa Barb. Rodr. | Paraboloid |
| Pinus sp. | Paraboloid |
| Pouteria caimito (Ruiz & Pav.) Radlk. | Cone |
| Tabebuia rosea (Bertol.) A. DC. | Cone |
| Anacardium excelsum (Kunth) Skeels | Cone |
| Symphonia globulifera L. f. | Cone |
| Dialyanthera acuminata Standl. | Cone |
| Otoba sp. | Cone |
The estimation method by dendrometric bodies is used as a basis for research work, since it provides accurate information; however, most of the time the rigorous volume estimation method is used because it is easier (Melo & Lizarazo, 2017; Romahn de la Vega & Ramírez-Maldonado, 2010).
Rigorous volumen estimation
Volume of stem and branch sections in tropical species are frequently estimated with the Smalian, Newton and Huber equations, expressed respectively with the following formulas:
where,
V |
volume (m3) |
gi |
sectional area at one end of the section (m2) |
gs+1 |
sectional area at the other end of the section (m2) |
gm |
sectional area taken in the middle of the section (m2) |
L |
length (m) |
e |
model error. |
These equations are more accurate in volume estimations; furthermore, they are the basis for deriving the merchantable or total volume equations, which range from the simplest to the most complex ones (Biging, 1988; Fraver, Ringvall, & Jonsson, 2007; Tlaxcala-Méndez, De los Santos-Posadas, Hernández-De la Rosa, & López-Ayala, 2016).
The accuracy of the equations depends on the morphological characteristics of the species, since sometimes Smalian's formula provides better estimates (Machado, Pereira, & Ríos, 2003; Marques da Silva-Binoti, Breda-Binoti, Gleriani, & García-Leite, 2009) and sometimes Newton's equation is usually more accurate than Smalian's (Fraver et al., 2007). This can be seen in the work of Akindele and LeMay (2006), who obtained the best volume estimates of tropical trees in Nigeria, Africa, using Newton's formula. Atha, Romero, and Forrest (2005) found no notable differences in estimates with the Smalian and Newton methods, while Biging (1988) had better results with the Newton and Huber equations.
Graphical method
This technique involves measuring diameters at different heights to determine the corresponding cross-sectional areas. These areas are plotted based on the height of volume estimation and a curve joining the points is drawn; subsequently, the curve is measured using a planimeter or computer programs, in order to quantify the area occupied by the solid (Aldana-Pereira, 2008), given by the following formula:
where,
V |
volume (m3) |
K |
transformation or scale coefficient |
di |
diameter at a height (m) |
e |
model error. |
Prodan (1961) considers that graphical methods provided the guideline to calculate the regressions analytically and, consequently, the development of the current statistical methods, where the analysis and interpretation of the regression are simpler through the use of computer systems. Logarithmic and arithmetic models can be described from the regressions.
Conventional equation systems
Equation systems allow estimating the volume of segments of interest of the tree, such as total stem volume, branch volume or total tree volume outside and inside bark, which can be at individual tree or stand level (Kitikidou, Milios, & Katsogridakis, 2017). The systems are based on regression models (Atha et al., 2005; Callister, England, & Collins, 2013; Da Cunha & Guimarães-Finger, 2009; Gómez-González et al., 2018; Goussanou et al., 2016; Machado et al., 2003; Mayaka, Eba'a-Atyi, & Momo, 2017) known as conventional equations; these can be linear, non-linear or logarithmic, among others. Non-linear equations are the most effective, as they best describe volume trends (Corral-Rivas & Návar-Chaidez, 2009). In the papers reviewed, it was found that the Schumacher and Hall (1933) equation, followed by Spurr’s (1952), are frequently used in tropical species since they adequately represent the estimated data sets with respect to observed ones (Crecente-Campo, Corral-Rivas, Vargas-Larreta, & Wehenkel, 2014). The models for estimating total tree volume are expressed as follows:
Schumacher and Hall
Spurr
Cone
where,
V |
volume (m3) |
d |
diameter at breast height (cm) at 1.30 m above ground level |
h |
total height (m) |
bi |
parameters to be estimated |
e |
model error. |
Through simultaneous equations it is possible to estimate the total volume, by mathematically integrating a volume equation and the cone formula, for scaling the branches (Fang & Bailey, 2001; Özçelik & Göçeri, 2015):
Several authors agree that the Schumacher and Hall (1933) function guarantees accurate estimates for different species (Cruz-Cobos, De los Santos-Posadas, & Valdez-Lazalde, 2008; Parresol, Hotvedt, & Cao, 1987; Vargas-Larreta et al., 2017), through ordinary least squares regression fitting techniques and the apparently uncorrelated regression technique, the latter providing significant gains in parameter optimization (Tamarit-Urias, 2013).
Similarly, traditional volume-taper equations have been used, from the most basic (Kozak, Munro, & Smith, 1969) to the most complex, such as segmented models which involve integrating merchantable volume and relative diameter for tropical species (López, Barrios, & Trincado, 2015; Özçelik & Göçeri, 2015; Niño-López, Ramos-Molina, Barrios, & López-Aguirre, 2018).
The Fang, Borders, and Bailey (2000) equation has also been used to estimate volume:
where,
Vc |
merchantable volume (m3) |
h |
total height (m) |
hb |
stump height (m) |
k |
π/40000 |
q |
hi/h |
αi, bi, pi |
parameters to be estimated |
e |
model error. |
All volume models improve in the estimation of their parameters with the mixed effects fitting technique, adding a bivariate formula with fixed and random effects, which allows accurately describing the stem of individual trees (Bueno-López & Bevilacqua, 2012; Kitikidou et al., 2017). This avoids any anomaly of correlation, multicollinearity and inflation of variances, surpassing previous methods in predictive ability (Fang & Bailey, 2001).
Because of this, Melo and Lizarazo (2017) generated a polynomial-type model as an alternative means of estimating both the merchantable and total volume of the standing tree, at different sections and that is flexible enough to model the volume of individual trees and stands:
where,
p (x) |
= polynomial function |
d |
diameter at breast height (cm) at 1.30 m above ground level |
h |
total tree height (m) |
di |
diameter at a height ih/10 (cm) |
ih/10 |
number of sections drawn on the tree (m) |
e |
model error. |
There is another technique for estimating total volume known as volume ratio function, developed by Burkhart (1977), which allows calculating the volume for different values of diameter or height defined in a simpler way than the taper models for tropical species (Barrios, López, & Nieto, 2014; Gilabert & Paci, 2010; Pece, 1994). Ratio functions provide significant gains to parameters (Hernández-Ramos et al., 2017). Due to the conditions of the growth sites, competition, and other environmental factors, a tree with greater taper has less volume than one with lesser taper having the same diameter at breast height (1.30 m) (Kitikidou et al., 2017). Using these expressions, it is possible to generate volume rates or volume tables of one input, two inputs (Guerra, Soudre, & Chota, 2008) and three inputs (diameter at breast height, height and form factor) for various species, preferably commercial (Aldana-Pereira, 2008), although the three-input ones are not frequently used.
The Cao, Burkhart, and Max (1980) and Burkhart (1977) equations have been frequently used and have provided good estimates for Eucalyptus grandis W. Mill ex Maiden, E. pellita F. Muell., E. urophylla S. T. Blake, E. nitens (H. Deane & Maiden) Maiden, Swietenia macrophylla King and Avicennia germinans (L.) L., among others (Hernández-Ramos et al., 2017, 2018). These equations are expressed respectively as follows:
Artificial neural networks
The artificial neural network method allows accurately estimating the total volume through diameter, total height, diameters at different heights and volume in different sections. The method provides reliable estimates as well as linear regression techniques, specifically the Schumacher and Hall (1933) equation that has been used in several studies (Marques da Silva-Binoti et al., 2009; Mena-Frau & Montecinos-Guajardo, 2006); however, in some studies, the Levenberng-Marquardt neural models have outperformed nonlinear equations in terms of mean error and fit index (Özçelik, Diamantopoulou, & Brooks, 2014). The advantage of the method is that it requires few measurements of the diameter at different heights of representative trees, which consequently reduces costs without losing accuracy in estimating the merchantable or total volume outside and inside bark (Marques da Silva-Binoti et al., 2014).
Criteria for the selection of the analyzed techniques
Based on the review, the accuracy of techniques such as the xylometer, Pressler and standing tree volume procedures, as well as dendrometric bodies, rigorous volume estimation and the graphical method, depends on the good use of equipment in the field work, the species, geometric shape of the tree and the personnel trained to analyze the data; these are criteria that guarantee the quality of the information. In the conventional and neural network equations, most researchers selected the best models using the following criteria: the coefficient of determination (R2) or adjusted coefficient of determination (R2adj), which should be close to 1, and the root mean square error (RMSE), which should tend to zero (Guzmán-Santiago et al., 2019). On some occasions they also considered bias, which holds that the average of the residuals should be equal to zero, thus making the estimator centered or unbiased; the Durbin-Watson test, which should be equal to 2 to indicate that there is no autocorrelation; and the Akaike and Bayesian information criteria that allow choosing the model with the lowest value (Del Carmen-García, Castellana, Rapelli, Koegel, & Catalano, 2014).
To reaffirm the selection of the models, some authors prefer to make a graphical analysis of the residuals against the predicted values, since this procedure is considered one of the most efficient ways of evaluating the fit capacity of a model (Martínez-Rodríguez, 2005).
Conclusions
The structures of the equations for estimating volume range from the simplest to the most complex. In this case, since we are dealing with biological material (trees), it is not possible to generate the data of interest accurately in a regression, so there are no better models than others, since each one is fitted to the researcher’s needs. Nevertheless, the xylometer technique is considered the most accurate to estimate volume, since it is used directly; however, the application of this technique is difficult in large logs. On the other hand, the rigorous volume estimation technique is the most used in the tropics to obtain the volume of the individual sections, which are fitted by the conventional or neural network equations to estimate the total volume; the latter generate slightly superior results.
