# Developing Allometric Equations for Teak Plantations Located in the Coastal Region of Ecuador from Terrestrial Laser Scanning Data

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}× h, by applying a robust regression method to remove likely outliers. Results showed that the developed allometric models performed reasonably well, especially those based on the metric DBH

^{2}× h, providing low bias estimates and relative RMSE values of 21.60% and 16.41% for TSCV and TSV, respectively. Allometric models only based on tree height were derived from replacing DBH by h in the expression DBH

^{2}x h, according to adjusted expressions depending on DBH classes (ranges of DBH). This finding can facilitate the obtaining of variables such as AGB (carbon stock) and commercial volume of wood over teak plantations in the Coastal Region of Ecuador from only knowing the tree height, constituting a promising method to address large-scale teak plantations monitoring from the canopy height models derived from digital aerial stereophotogrammetry.

## 1. Introduction

^{3}/year [6].

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Field Data

^{2}/ha (from 0.6 to 6.28 m

^{2}/ha) and a low average Lorey’s mean height of 7.83 m (from 3.84 to 9.57 m). The average slope of the reference plots was 23.4%, ranging from 3.9% to 55.5% [34]. Understory vegetation was mainly composed of shrubs (Mimosa pudica Linn.) and an herbaceous layer (Convolvulus arvensis Linn.). However, ground level was detectable since herbaceous cover was relatively transparent at the time of field work. Clearing was needed to remove shrubs and some herbaceous cover before carrying out the TLS field work.

^{2}/ha and 15.36 m, respectively, significantly higher than those registered in Morondava. Specifically, the average basal area increment with respect to Morondava was 319.5%. Considering that both basal area [38] and Lorey’s mean height [39] are dasometric variables positively correlated to forest stand volume, it can be inferred that the reference plots located in El Tecal presented a forest stand volume larger than those of Morondava. The average slope of the reference plots was 11.7%, ranging from 8.3% to 14.7% [34]. Understory vegetation was mainly composed of an herbaceous layer (Amaranthus retroflexus L. and different species of grasses). Shrubs were scarce due to high competition with teak trees. Ground level was practically covered with fallen teak leaves.

^{2}/ha (11.56 m

^{2}/ha on average, meaning an average basal area increment with respect to Morondava of 189%), while Lorey’s mean height values varied from 12.30 to 23.02 m (18.57 m on average). The average slope of the reference plots was 16.8%, ranging from 2.5% to 43.9% [34]. Understory vegetation was composed of a dense herbaceous layer (grasses such as Rottboellia exaltata L. and Panicum maximum L.) and a scarcely developed shrub layer due to competition with teak trees. Ground level was entirely covered with fallen teak leaves. Clearing was needed to remove herbaceous cover before TLS field work.

^{©}7.1 software (FARO Technologies Inc., Lake Mary, FL, USA) was utilized to co-register the four scans within each reference plot, thus producing a single and colored 3D point cloud, from using nine artificial targets (15 cm diameter spheres) conveniently distributed over the reference plot to ensure that at least three spheres were visible from every two consecutive scan positions. The mean distance error of the co-registered point clouds for the 58 reference plots was 6.2 mm, with maximum and minimum values of 13.3 and 2 mm, respectively.

^{2}(28,472 on average). Point density ranged from 28,090 to 33,868 points/m

^{2}(31,859 on average) in El Tecal plantation, while it varied between 29,472 and 63,038 points/m

^{2}(40,820 on average) in Allteak plantation.

#### 2.3. Tree Database Collection

#### 2.3.1. Automated Tree Segmentation

#### 2.3.2. Selection of Suitable Trees

#### 2.4. Automatic Extraction of Tree Level Information

_{i}) (normalized heights with respect to the ground), starting with two sections at heights of 0.15 and 0.30 m. From the height of 0.30 m, sections are produced according to increments of 0.50 m, thus including in the dataset the value of 1.30 m where the DBH is measured. Each cross section h

_{i}encompasses points from the tree stem TLS point cloud located at a height h

_{i}± 2.5 cm. The section is considered valid if the number of points is greater than 20. This means that there may be some missing cross sections because they have few or no TLS points due to occlusion problems during the scanning process. The points included in each cross section i, after their 2D projection onto the horizontal plane, are adjusted to a circle using the robust adjustment method proposed by Ladrón de Guevara et al. [44]. This method of adjustment is very suitable when outliers are foreseen due to TLS points not belonging to the tree stem (e.g., those located in epiphytic vegetation, nearby branches, or understory vegetation). It is also recommended to deal with the adjustment of incomplete circles (circle arcs), as can be the case in TLS inventories when the entire circle of a tree stem cross section is not covered because of occlusion problems or simply due to an insufficient number of scanning positions in the reference plot [19]. Since most geometric or algebraic adjustment methods are sensitive to noise and atypical points, Tree_geometry uses the minimum absolute error (MAE) criterion to provide robust estimates. Considering that the MAE objective function is not differentiable, and so the adjustment methods based on the gradient cannot be applied, the implemented algorithm relies on the partial derivative on the right and on the left (exterior and interior points to the reference circle in each iteration). It allows to significantly increase its computational efficiency, representing a good alternative to those methods based on least squares fitting, RANSAC-based least squares fitting, or randomized Hough transform [18,22,41].

_{i,i}

_{+1}between two consecutive cross sections S

_{i}and S

_{i}

_{+1}of radii R

_{i}and R

_{i}

_{+1}) was computed as the volume of a truncated cone given by the following expression:

_{i,i}

_{+1}being the difference in height between the two consecutive cross sections S

_{i}and S

_{i}

_{+1}(usually 0.5 m). The sum of the volumes of all the logs, including the final one up to the diameter of 0.13 m, would be a good estimation of the commercial stem volume of each tree.

#### 2.5. Development and Validation of Allometric Models

_{i}represents the value of the normalized residue for the i-th point which, in turn, is given by the expressions shown in Equation 8.

_{r}refers to a value of dispersion of the residuals given by the deviation of the absolute values of the residuals with respect to the median (MAD) divided by 0.6745 (S

_{r}= MAD/0.6745). On the other hand, h

_{i}is the i-th value of the diagonal of the matrix [A(A

^{t}A

^{)-1}A

^{t}] (least squares adjustment), where A is the matrix of coefficients of the multiple linear regression according to the model Y = AX + ε.

_{pred}and Y

_{obs}refers to the predicted and observed values, respectively.

_{pred}= e

^{(α+βln(DBH)+ε)}. As it was said before, ε is assumed to be normally distributed [i.e., N(0, σ

^{2})], therefore the mean of ${e}^{\epsilon}$ can be approximated by ${e}^{\frac{{\sigma}^{2}}{2}}$ [52]. Note that the term ${e}^{\frac{{\sigma}^{2}}{2}}$ can be understood as a correction factor applied to back-transform predicted values and remove bias from the log-transformed data [28,52]. An unbiased estimate of Y

_{pred}can therefore be calculated using the following equation:

- Global fitting. The model parameters were computed from all available sample teak trees (2272 individuals).
- DBH-based fitting. The original sample of teak trees was grouped into three classes according to DBH ranges ((5, 10), (10, 20), and (20, 30) cm). Therefore, three groups of model parameters were adjusted, one for each DBH range.

_{i}refers to the set of n observations (residuals in our case), M is the median of the observations and k is a constant scale factor, which depends on the distribution. Usually k = 1.4826 if normality of the data is assumed, disregarding the abnormality induced by outliers [57].

## 3. Results

#### 3.1. Relationship Between DBH and Total Tree Height

#### 3.2. Allometric Model to Estimate Tree Commercial Volume

^{2}× h (Equation (5)), according to the DBH-based fitting model previously described.

^{3}, which implied an estimation error (relative RMSE) of between 21.6% and 33.22%. The two-variable model, which had DBH and h as explanatory variables, turned out to be the most accurate, with relative RMSE and RMSE values of 21.6% and 0.0482 m

^{3}, respectively. The allometric model based only on tree height, an attractive model because tree height is relatively easy to obtain by means of UAV-based stereo-photogrammetry or airborne lidar [12,58], presented an estimation error of 33.22% (RMSE = 0.074 m

^{3}), so it would allow a reasonable approximation for the estimation of the commercial volume of teak plantations.

^{3}.

^{3}. Despite these shortcomings, when robust statistics insensitive to outliers such as median (central tendency) and MAD (variability) were computed over the residuals, reasonably low values of −0.0103 and 0.0407 m

^{3}were obtained, respectively. It should be highlighted that, in this case, the allometric model only depends on tree height, a variable that can be remotely sensed from spaceborne or airborne sensors.

#### 3.3. Allometric Model to Estimate Tree Stem Volume and Dry Biomass

^{2}× h) is highlighted since it performed predictions with a practically negligible bias (overestimation of 0.37%), a low RMSE of 0.0319 m

^{3}, and a relative estimation error of only 16.41%. The allometric model based on only tree height performed slightly worse, presenting a bias of 2.54% and RMSE and relative estimation error of 0.0503 m

^{3}and 25.88%, respectively.

^{3}, which corresponded with a relative estimation error of 23.72%. In this sense, this derived model would allow predicting tree stem volume with an average error of less than 25% in teak plantations only from knowing tree height.

^{3}. As in the case of commercial volume, there was a tendency to underpredicting tree stem volume at the largest tree sizes, that is tree stem volumes above 0.5 m

^{3}. In addition, it is worth noting the likely presence of outliers in the dataset since very low values of −0.002 and 0.0247 m

^{3}were obtained when computing robust statistics such as median and MAD, respectively.

_{stem}is given in m

^{3}, and DBH and h are expressed in cm and m, respectively. The fit of this model to the sample data collected in this work is shown in Figure 6b, pointing to a clear underestimation of the actual values even for medium size trees. In quantitative terms, the bias of the adjustment was −6.80%, presenting a RMSE value of 0.0521 m

^{3}and a relative RMSE of 26.79%, so performing worse than any of the allometric models developed in this investigation (see Table 5). This performance difference was especially notable in the case of the allometric model based on the metric DBH

^{2}× h, a comparable model because it uses the same input variables, that provided unbiased results (bias = 0.37%) and low RMSE and relative RMSE values of 0.0319 m

^{3}and 16.41%, respectively.

^{3}, an average value extracted from bibliography for areas such as Ecuador, Brazil, Venezuela, Bolivia, and Costa Rica [60,61,62,63,64]. It should be borne in mind that the density of teak wood is relatively variable with the age of the tree, increasing with increasing age [62].

#### 3.4. Allometric Model to Estimate the Above-Ground Biomass at Tree Level

- If we have the DBH and h of each tree:$$\mathrm{AGB}\left(\frac{\mathrm{kg}}{\mathrm{tree}}\right)=\frac{550}{0.8}.{e}^{\left(-9.5344+0.9334.\mathrm{ln}\left({\mathrm{DBH}}^{2}\mathrm{h}\right)\right)}{e}^{\frac{{0.1373}^{2}}{2}}.$$
- If we only have the tree height:Equation (15) would be applied, but substituting DBH for its estimation from the following expressions (see the DBH-based fitting model parameters in Table 2):$$\mathrm{DBH}={e}^{\left(0.8904+0.5696.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.1167}^{2}}{2}}\left(5\mathrm{cm}\le \mathrm{DBH}10\mathrm{cm}\right)$$$$\mathrm{DBH}={e}^{\left(0.5605+0.7863.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.1094}^{2}}{2}}\left(10\mathrm{cm}\le \mathrm{DBH}20\mathrm{cm}\right)$$$$\mathrm{DBH}={e}^{\left(2.143+0.3148.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.0685}^{2}}{2}}\left(20\mathrm{cm}\le \mathrm{DBH}30\mathrm{cm}\right)$$

#### 3.5. Descriptive Statistics of Some Dendrometric Variables

^{3}on average. Similarly, Mora and Hernández [66], working on teak plantations in the Pacific of Costa Rica, reported an average commercial volume of 0.48 m

^{3}for the DBH class between 25 and 30 cm, a value that fully coincides with that obtained in the present investigation.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kollert, W.; Kleine, M. (Eds.) The Global Teak Study: Analysis, Evaluation and Future Potential of Teak Resources; IUFRO World Series Volume 36: Vienna, Austria, 2017; ISBN 978-3-902762-77-1. [Google Scholar]
- Food and Agriculture Organization of the United Nations. The State of the World’s Forest Genetic Resources; FAO of the United Nations: Rome, Italy, 2014. [Google Scholar]
- Kollert, W.; Cherubini, L. Teak Resources and Market. Assessment 2010; Planted Forests and Trees Working Paper FP/47/E; FAO of the United Nations: Rome, Italy, 2012. [Google Scholar]
- Midgley, S.; Somaiya, R.T.; Stevens, P.R.; Brown, A.; Nguyen, D.K.; Laity, R. Planted Teak: Global Production and Markets, with Reference to Solomon Islands; Technical Report 85; Australian Center for International Agricultural Research: Canberra, Australia, 2015.
- Cañadas, Á.; Andrade-Candell, J.; Domínguez, J.M.; Molina, C.; Schnabel, O.; Vargas-Hernández, J.J.; Wehenkel, C. Growth and Yield Models for Teak Planted as Living Fences in Coastal Ecuador. Forests
**2018**, 9, 55. [Google Scholar] [CrossRef] - Kollert, W.; Przemyslaw, J.W. (Eds.) World Teak Resources, Production, Markets and Trade. In The Global Teak Study. Analysis, Evaluation and Future Potential of Teak Resources; IUFRO World Series Volume 36: Vienna, Austria, 2017; pp. 83–89. ISBN 978-3-902762-77-1. [Google Scholar]
- Agrawal, A.; Nepstad, D.; Chhatre, A. Reducing Emissions from Deforestation and Forest Degradation. Annu. Rev. Environ. Resour.
**2011**, 36, 373–396. [Google Scholar] [CrossRef] - Dixon, R.K.; Solomon, A.M.; Brown, S.; Houghton, R.A.; Trexier, M.C.; Wisniewski, J. Carbon pools and flux of global forest ecosystems. Science
**1994**, 263, 185–190. [Google Scholar] [CrossRef] [PubMed] - Houghton, R.A.; Nassikas, A.A. Negative emissions from stopping deforestation and forest degradation, globally. Glob. Chang. Biol.
**2018**, 24, 350–359. [Google Scholar] [CrossRef] [PubMed] - Herold, M.; Johns, T. Linking requirements with capabilities for deforestation monitoring in the context of the UNFCCC-REDD process. Environ. Res. Lett.
**2007**, 2, 1–7. [Google Scholar] [CrossRef] - Gibbs, H.K.; Brown, S.; Niles, J.O.; Foley, J.A. Monitoring and estimating tropical forest carbon stocks: Making REDD a reality. Environ. Res. Lett.
**2007**, 2, 045023. [Google Scholar] [CrossRef] - Ferraz, A.; Saatchi, S.; Mallet, C.; Meyer, V. Lidar detection of individual tree size in tropical forests. Remote Sens. Environ.
**2016**, 183, 318–333. [Google Scholar] [CrossRef] - Chave, J.; Réjou-Méchain, M.; Búrquez, A.; Chidumayo, E.; Colgan, M.S.; Delitti, W.B.C.; Duque, A.; Eid, T.; Fearnside, P.M.; Goodman, R.C.; et al. Improved allometric models to estimate the aboveground biomass of tropical trees. Glob. Chang. Biol.
**2014**, 20, 3177–3190. [Google Scholar] [CrossRef] - Jucker, T.; Caspersen, J.; Chave, J.; Antin, C.; Barbier, N.; Bongers, F.; Dalponte, M.; van Ewijk, K.Y.; Forrester, D.I.; Haeni, M.; et al. Allometric equations for integrating remote sensing imagery into forest monitoring programmes. Glob. Chang. Biol.
**2017**, 23, 177–190. [Google Scholar] [CrossRef] - Wulder, M.A.; Franklin, S.E. (Eds.) Remote Sensing of Forest Environments, Concepts and Case Studies, 1st ed.; Kluwer Academic Publishers: Boston, MA, USA, 2003; ISBN 978-1-4020-7405-9. [Google Scholar]
- White, J.C.; Coops, N.C.; Wulder, M.A.; Vastaranta, M.; Hilker, T.; Tompalski, P. Remote Sensing Technologies for Enhancing Forest Inventories: A Review. Can. J. Remote Sens.
**2016**, 42, 619–641. [Google Scholar] [CrossRef] - Giannetti, F.; Puletti, N.; Quatrini, V.; Travaglini, D.; Bottalico, F.; Corona, P.; Chirici, G. Integrating terrestrial and airborne laser scanning for the assessment of single-tree attributes in Mediterranean forest stands. Eur. J. Remote Sens.
**2018**, 51, 795–807. [Google Scholar] [CrossRef] - Suraj Reddy, R.; Rakesh, A.; Jha, C.S.; Rajan, K.S. Automatic Estimation of Tree Stem Attributes Using Terrestrial Laser Scanning in Central Indian Dry Deciduous Forests. Curr. Sci.
**2018**, 114, 201–206. [Google Scholar] [CrossRef] - Liang, X.; Kankare, V.; Hyyppä, J.; Wang, Y.; Kukko, A.; Haggrén, H.; Yu, X.; Kaartinen, H.; Jaakkola, A.; Guan, F.; et al. Terrestrial laser scanning in forest inventories. ISPRS J. Photogramm. Remote Sens.
**2016**, 115, 63–77. [Google Scholar] [CrossRef] - Du, S.; Lindenbergh, R.; Ledoux, H.; Stoter, J.; Nan, L. AdTree: Accurate, Detailed, and Automatic Modelling of Laser-Scanned Trees. Remote Sens.
**2019**, 11, 2074. [Google Scholar] [CrossRef] - Saarinen, N.; Kankare, V.; Vastaranta, M.; Luoma, V.; Pyörälä, J.; Tanhuanpää, T.; Liang, X.; Kaartinen, H.; Kukko, A.; Jaakkola, A.; et al. Feasibility of Terrestrial laser scanning for collecting stem volume information from single trees. ISPRS J. Photogramm. Remote Sens.
**2017**, 123, 140–158. [Google Scholar] [CrossRef] - Maas, H.-G.; Bienert, A.; Scheller, S.; Keane, E. Automatic forest inventory parameter determination from terrestrial laser scanner data. Int. J. Remote Sens.
**2008**, 29, 1579–1593. [Google Scholar] [CrossRef] - Dassot, M.; Constant, T.; Fournier, M. The use of terrestrial LiDAR technology in forest science: Application fields, benefits and challenges. Ann. For. Sci.
**2011**, 68, 959–974. [Google Scholar] [CrossRef] - Liang, X.; Kankare, V.; Yu, X.; Hyyppä, J.; Holopainen, M. Automated Stem Curve Measurement Using Terrestrial Laser Scanning. IEEE Trans. Geosci. Remote Sens.
**2014**, 52, 1739–1748. [Google Scholar] [CrossRef] - Holopainen, M.; Vastaranta, M.; Kankare, V.; Räty, M.; Vaaja, M.; Liang, X.; Yu, X.; Hyyppä, J.; Hyyppä, H.; Viitala, R.; et al. Biomass estimation of individual trees using stem and crown diameter TLS measurements. ISPRS Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2012**, XXXVIII-5/W12, 91–95. [Google Scholar] [CrossRef] - Kankare, V.; Holopainen, M.; Vastaranta, M.; Puttonen, E.; Yu, X.; Hyyppä, J.; Vaaja, M.; Hyyppä, H.; Alho, P. Individual tree biomass estimation using terrestrial laser scanning. ISPRS J. Photogramm. Remote Sens.
**2013**, 75, 64–75. [Google Scholar] [CrossRef] - Calders, K.; Newnham, G.; Burt, A.; Murphy, S.; Raumonen, P.; Herold, M.; Culvenor, D.; Avitabile, V.; Disney, M.; Armston, J.; et al. Nondestructive estimates of above-ground biomass using terrestrial laser scanning. Methods Ecol. Evol.
**2015**, 6, 198–208. [Google Scholar] [CrossRef] - Lau, A.; Calders, K.; Bartholomeus, H.; Martius, C.; Raumonen, P.; Herold, M.; Vicari, M.; Sukhdeo, H.; Singh, J.; Goodman, R. Tree Biomass Equations from Terrestrial LiDAR: A Case Study in Guyana. Forests
**2019**, 10, 527. [Google Scholar] [CrossRef] - Raumonen, P.; Kaasalainen, M.; Åkerblom, M.; Kaasalainen, S.; Kaartinen, H.; Vastaranta, M.; Holopainen, M.; Disney, M.; Lewis, P. Fast Automatic Precision Tree Models from Terrestrial Laser Scanner Data. Remote Sens.
**2013**, 5, 491–520. [Google Scholar] [CrossRef] - Lau, A.; Bentley, L.P.; Martius, C.; Shenkin, A.; Bartholomeus, H.; Raumonen, P.; Malhi, Y.; Jackson, T.; Herold, M. Quantifying branch architecture of tropical trees using terrestrial LiDAR and 3D modelling. Trees
**2018**, 32, 1219–1231. [Google Scholar] [CrossRef] - Hackenberg, J.; Spiecker, H.; Calders, K.; Disney, M.; Raumonen, P. SimpleTree—An Efficient Open Source Tool to Build Tree Models from TLS Clouds. Forests
**2015**, 6, 4245–4294. [Google Scholar] [CrossRef] - Delagrange, S.; Jauvin, C.; Rochon, P. PypeTree: A Tool for Reconstructing Tree Perennial Tissues from Point Clouds. Sensors
**2014**, 14, 4271–4289. [Google Scholar] [CrossRef] - Gonzalez de Tanago, J.; Lau, A.; Bartholomeus, H.; Herold, M.; Avitabile, V.; Raumonen, P.; Martius, C.; Goodman, R.C.; Disney, M.; Manuri, S.; et al. Estimation of above-ground biomass of large tropical trees with terrestrial LiDAR. Methods Ecol. Evol.
**2018**, 9, 223–234. [Google Scholar] [CrossRef] - Aguilar, F.J.; Rivas, J.R.; Nemmaoui, A.; Peñalver, A.; Aguilar, M.A.; Aguilar, F.J.; Rivas, J.R.; Nemmaoui, A.; Peñalver, A.; Aguilar, M.A. UAV-Based Digital Terrain Model Generation under Leaf-Off Conditions to Support Teak Plantations Inventories in Tropical Dry Forests. A Case of the Coastal Region of Ecuador. Sensors
**2019**, 19, 1934. [Google Scholar] [CrossRef] - Flores Velasteguí, T.; Cabezas Guerrero, F.; Crespo Gutiérrez, R. Plagas y enfermedades en plantaciones de Teca (Tectona grandis L.F.) en la zona de Balzar, provincia de Guayas. Cienc. Tecnol.
**2010**, 3, 15–22. [Google Scholar] [CrossRef] - Holdridge, L.R. Ecología Basada en Zonas de Vida; Instituto Interamericano de Cooperacion para la Agricultura: San José, Costa Rica, 1982; ISBN 9789290390398. [Google Scholar]
- Puletti, N.; Grotti, M.; Scotti, R. Evaluating the eccentricities of poplar stem profiles with terrestrial laser scanning. Forests
**2019**, 10, 239. [Google Scholar] [CrossRef] - Chen, Q.; Gong, P.; Baldocchi, D.; Tian, Y.Q. Estimating Basal Area and Stem Volume for Individual Trees from Lidar Data. Photogramm. Eng. Remote Sens.
**2007**, 73, 1355–1365. [Google Scholar] [CrossRef] - Tran-Ha, M.; Cordonnier, T.; Vallet, P.; Lombart, T. Estimation du volume total aérien des peuplements forestiers à partir de la surface terrière et de la hauteur de Lorey. Rev. For. Fr.
**2011**, 63, 361–378. [Google Scholar] [CrossRef][Green Version] - Pueschel, P.; Newnham, G.; Rock, G.; Udelhoven, T.; Werner, W.; Hill, J. The influence of scan mode and circle fitting on tree stem detection, stem diameter and volume extraction from terrestrial laser scans. ISPRS J. Photogramm. Remote Sens.
**2013**, 77, 44–56. [Google Scholar] [CrossRef] - Trochta, J.; Krůček, M.; Vrška, T.; Král, K. 3D Forest: An application for descriptions of three-dimensional forest structures using terrestrial LiDAR. PLoS ONE
**2017**, 12, e0176871. [Google Scholar] [CrossRef][Green Version] - Tao, S.; Wu, F.; Guo, Q.; Wang, Y.; Li, W.; Xue, B.; Hu, X.; Li, P.; Tian, D.; Li, C.; et al. Segmenting tree crowns from terrestrial and mobile LiDAR data by exploring ecological theories. ISPRS J. Photogramm. Remote Sens.
**2015**, 110, 66–76. [Google Scholar] [CrossRef][Green Version] - Zhang, W.; Wan, P.; Wang, T.; Cai, S.; Chen, Y.; Jin, X.; Yan, G. A novel approach for the detection of standing tree stems from plot-level terrestrial laser scanning data. Remote Sens.
**2019**, 11, 211. [Google Scholar] [CrossRef][Green Version] - Ladrón de Guevara, I.; Muñoz, J.; de Cózar, O.D.; Blázquez, E.B. Robust Fitting of Circle Arcs. J. Math. Imaging Vis.
**2011**, 40, 147–161. [Google Scholar] [CrossRef] - Telles, R.; Gómez, M.; Alanís, E.; Aguirre, O.A.; Jiménez, J. Ajuste y selección de modelos matemáticos para predecir el volumen total fustal de Tectona grandis en Nuevo Urecho, Michoacán, México. Madera Bosques
**2018**, 24, e2431544. [Google Scholar] - Bermejo, I.; Cañellas, I.; Miguel, A.S. Growth and yield models for teak plantations in Costa Rica. For. Ecol. Manag.
**2004**, 189, 97–110. [Google Scholar] [CrossRef] - Armijos Guzmán, D.D. Construcción de Tablas Volumétricas y Cálculo del Factor de Forma para dos Especies, Teca (Tectona Grandis) y Melina (Gmelina arborea), en Tres Plantaciones de la Empresa Reybanpac C.A. en la Provincia de Los Ríos. Ph.D. Thesis, Escuela Superior Politécnica de Chimborazo, Escuela de Ingeniería Forestal, Cartago, Costa Rica, 2013. [Google Scholar]
- Ounban, W.; Puangchit, L.; Diloksumpun, S. Development of general biomass allometric equations for Tectona grandis Linn.F. and Eucalyptus camaldulensis Dehnh. plantations in Thailand. Agric. Nat. Resour.
**2016**, 50, 48–53. [Google Scholar] [CrossRef][Green Version] - Pérez, L.D.; Kanninen, M. Aboveground biomass of Tectona grandis plantations in Costa Rica. J. Trop. For. Sci.
**2003**, 15, 199–213. [Google Scholar] - Dumouchel, W.; O’Brien, F. Integrating a robust option into a multiple regression computing environment. Inst. Math. Its Appl.
**1991**, 36, 41. [Google Scholar] - Street, J.O.; Carroll, R.J.; Ruppert, D. A Note on Computing Robust Regression Estimates Via Iteratively Reweighted Least Squares. Am. Stat.
**1988**, 42, 152. [Google Scholar] - Baskerville, G.L. Use of Logarithmic Regression in the Estimation of Plant Biomass. Can. J. For. Res.
**1972**, 2, 49–53. [Google Scholar] [CrossRef] - Sumida, A.; Miyaura, T.; Torii, H. Relationships of tree height and diameter at breast height revisited: Analyses of stem growth using 20-year data of an even-aged Chamaecyparis obtusa stand. Tree Physiol.
**2013**, 33, 106–118. [Google Scholar] [CrossRef] [PubMed] - Iizuka, K.; Yonehara, T.; Itoh, M.; Kosugi, Y.; Iizuka, K.; Yonehara, T.; Itoh, M.; Kosugi, Y. Estimating Tree Height and Diameter at Breast Height (DBH) from Digital Surface Models and Orthophotos Obtained with an Unmanned Aerial System for a Japanese Cypress (Chamaecyparis obtusa) Forest. Remote Sens.
**2017**, 10, 13. [Google Scholar] [CrossRef][Green Version] - West, G.B.; Brown, J.H.; Enquist, B.J. A general model for the structure and allometry of plant vascular systems. Nature
**1999**, 400, 664–667. [Google Scholar] [CrossRef] - Huber, P.J.; Ronchetti, E.M. Robust Statistics, 2nd ed.; John Wiley & Sons: Hoboken, NJ, USA, 2009; ISBN 0470129905. [Google Scholar]
- Rousseeuw, P.J.; Croux, C. Alternatives to the Median Absolute Deviation. J. Am. Stat. Assoc.
**1993**, 88, 1273–1283. [Google Scholar] [CrossRef] - White, J.C.; Tompalski, P.; Coops, N.C.; Wulder, M.A. Comparison of airborne laser scanning and digital stereo imagery for characterizing forest canopy gaps in coastal temperate rainforests. Remote Sens. Environ.
**2018**, 208, 1–14. [Google Scholar] [CrossRef] - Lara, C.E. Aplicación de ecuaciones de conicidad para teca (Tectona grandis L.F.) en la zona costera ecuatoriana. Cienc. Tecnol.
**2012**, 4, 19–27. [Google Scholar] [CrossRef] - Crespo, R.; Jiménez, E.; Suatunce, P.; Law, G.; Sánchez, C. Análisis comparativo de las propiedades físico-mecánicas de la madera de teca (Tectona grandis L.F.) de Quevedo y Balzar. Cienc. Tecnol.
**2008**, 1, 55–63. [Google Scholar] [CrossRef] - Flórez, J.B.; Trugilho, P.F.; Lima, J.T.; Hein, P.R.G.; Silva, J.R.M. da Characterization of young wood Tectona grandis L. F. planted in Brazil. Madera Bosques
**2014**, 20, 11–20. [Google Scholar] - Valero, S.W.; Reyes, E.C.; Garay, D.A. Estudio de las propiedades físico-mecánicas de la especie Tectona grandis, de 20 años de edad, proveniente de las plantaciones de la Unidad Experimental de la Reserva Forestal Ticoporo, Estado Barinas. Rev. For. Venez.
**2005**, 49, 61–73. [Google Scholar] - Rivero, J.; Moya, R. Propiedades físico-mecánicas de la madera de Tectona grandis Linn. F. (teca), proveniente de una plantación de ocho años de edad en Cochabamba, Bolivia. Kurú Rev. For.
**2006**, 3, 1–14. [Google Scholar] - Pérez, L.D.; Kanninen, M. Heartwood, sapwood and bark content, and wood dry density of young and mature teak (Tectona grandis) trees grown in Costa Rica. Silva. Fenn.
**2003**, 37, 45–54. [Google Scholar] - Krishnapillay, D.B. Silviculture and management of teak plantations. Unasylva
**2000**, 51, 14–21. [Google Scholar] - Mora, F.; Hernández, W. Estimación del volumen comercial por producto para rodales de teca en el Pacífico de Costa Rica. Agron. Costarric.
**2007**, 31, 101–112. [Google Scholar] - Ota, T.; Ogawa, M.; Shimizu, K.; Kajisa, T.; Mizoue, N.; Yoshida, S.; Takao, G.; Hirata, Y.; Furuya, N.; Sano, T.; et al. Aboveground Biomass Estimation Using Structure from Motion Approach with Aerial Photographs in a Seasonal Tropical Forest. Forests
**2015**, 6, 3882–3898. [Google Scholar] [CrossRef][Green Version] - Guerra-Hernández, J.; González-Ferreiro, E.; Monleón, V.; Faias, S.; Tomé, M.; Díaz-Varela, R. Use of Multi-Temporal UAV-Derived Imagery for Estimating Individual Tree Growth in Pinus pinea Stands. Forests
**2017**, 8, 300. [Google Scholar] [CrossRef] - Popescu, S.C.; Wynne, R.H. Seeing the Trees in the Forest. Photogramm. Eng. Remote Sens.
**2004**, 70, 589–604. [Google Scholar] [CrossRef] - Feldpausch, T.R.; Lloyd, J.; Lewis, S.L.; Brienen, R.J.W.; Gloor, M.; Monteagudo Mendoza, A.; Lopez-Gonzalez, G.; Banin, L.; Abu Salim, K.; Affum-Baffoe, K.; et al. Tree height integrated into pantropical forest biomass estimates. Biogeosciences
**2012**, 9, 3381–3403. [Google Scholar] [CrossRef][Green Version] - Iida, Y.; Kohyama, T.S.; Kubo, T.; Kassim, A.R.; Poorter, L.; Sterck, F.; Potts, M.D. Tree architecture and life-history strategies across 200 co-occurring tropical tree species. Funct. Ecol.
**2011**, 25, 1260–1268. [Google Scholar] [CrossRef] - Hemery, G.E.; Savill, P.S.; Pryor, S.N. Applications of the crown diameter–stem diameter relationship for different species of broadleaved trees. For. Ecol. Manage.
**2005**, 215, 285–294. [Google Scholar] [CrossRef] - Spicer, R.; Groover, A. Evolution of development of vascular cambia and secondary growth. New Phytol.
**2010**, 186, 577–592. [Google Scholar] [CrossRef] [PubMed] - Shinozaki, K.; Yoda, K.; Hozumi, K.; Kira, T. A quantitative analysis of plant form. The pipe model theory: I. Basic analyses. Jpn. J. Ecol.
**1964**, 14, 97–105. [Google Scholar]

**Figure 1.**Situation map of the three teak plantations located in the province of Guayas (Ecuador): Morondava, El Tecal, and Allteak.

**Figure 2.**Reference plots geometry and Terrestrial Laser Scanning (TLS) derived point cloud. (

**a**) Circular reference plots for TLS field work showing the four scans pattern; (

**b**) semi-automatically segmented TLS point cloud showing ground (brown) and vegetation (green) classified points. Case study: Reference plot located in El Tecal plantation.

**Figure 4.**Example of a teak tree before and after manual editing. (

**a**): Point cloud representing the segmented tree before manual editing. (

**b**): The same segmented tree after manual editing from which the points have been manually classified in stem tree points (red color) and branches and leaves tree points (green color).

**Figure 5.**Plot of observed and predicted values for the allometric model given by the expression ${\mathrm{V}}_{com}={e}^{\left(-12.8904+1.2739.\mathrm{ln}\left(f{\left(h\right)}^{2}h\right)\right)}{e}^{\frac{{0.2746}^{2}}{2}}$, where f(h) represents the DBH-based fitting model which relates DBH and h. The red line refers to the 1:1 line.

**Figure 6.**Plot of observed and predicted values of tree stem volume. (

**a**) Allometric model given by the expression ${\mathrm{V}}_{stem}={e}^{\left(-9.5344+0.9334.\mathrm{ln}\left(f{\left(h\right)}^{2}h\right)\right)}{e}^{\frac{{0.1373}^{2}}{2}}$, where f(h) represents the DBH-based fitting model which relates DBH and h. (

**b**) Allometric model proposed by Lara [59]. The red line refers to the 1:1 line.

**Figure 7.**Plot of AGB values estimated from the model of Pérez and Kanninen (2003) [49] (horizontal axis) and the two AGB allometric modes proposed in this work (vertical axis):

**a**) Two-variables AGB model depending on DBH and tree total height;

**b**) AGB model based only on tree total height. The red line refers to the 1:1 line.

**Table 1.**Distribution of the teak trees selected to develop allometric models headed up to estimate both commercial and total stem volume.

Class | DBH (cm) | Selected Trees |
---|---|---|

1 | [5,10] | 100 |

2 | (10,15] | 100 |

3 | (15,20] | 146 |

4 | (20,25] | 100 |

5 | > 25 | 10 |

Total | 456 |

**Table 2.**Validation results of the robust regression to fit the model $DBH={e}^{\left(\propto +\beta \mathrm{ln}\left(h\right)\right)}{e}^{\frac{{\sigma}^{2}}{2}}$ from all available sample trees (Global fitting) and from sample trees grouped by DBH classes (DBH-based fitting). The standard deviation calculated on the 100 validation datasets is given in brackets.

Statistics | Global Fitting 2272 Trees | DBH-Based Fitting 5 ≤ DBH (cm) < 10 958 Trees | DBH-Based Fitting 10 ≤ DBH (cm) < 20 1110 Trees | DBH-Based Fitting 20 ≤ DBH (cm) < 30 204 Trees |
---|---|---|---|---|

RMSE (cm) | 1.69 (0.10) | 0.88 (0.07) | 1.53 (0.13) | 1.46 (0.10) |

Relative RMSE (%) | 13.85 | 10.65 | 11.10 | 6.65 |

Bias (%) | 1.32 (0.81) | 1.97 (1.03) | 0.81 (0.99) | 0.15 (0.62) |

σ | 0.1778 | 0.1167 | 0.1094 | 0.0685 |

α (p < 0.001) | 0.1055 | 0.8904 | 0.5605 | 2.1430 |

β (p < 0.001) | 0.9580 | 0.5696 | 0.7863 | 0.3148 |

**Table 3.**Error propagation when replacing DBH with the expression $DBH={e}^{\left(\propto +\beta \mathrm{ln}\left(h\right)\right)}{e}^{\frac{{\sigma}^{2}}{2}}$ for estimating AGB (above-ground biomass) values at tree level according to the allometric model proposed by Pérez and Kanninen [49].

Kind of Fitting Model | DBH Class * | RMSE (kg/tree) | Relative RMSE (%) | Bias (%) |
---|---|---|---|---|

Global fitting model $AGB=0.153{\left({e}^{\left(0.1055+0.958.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.1768}^{2}}{2}}\right)}^{2.382}$ | class 1 | 12.06 | 47.63 | 23.24 |

class 2 | 26.43 | 29.61 | 4.85 | |

class 3 | 71.14 | 28.55 | −19.84 | |

DBH-based fitting model ** $AGB=0.153{\left({e}^{\left(\propto +\beta \mathrm{ln}\left(h\right)\right)}{e}^{\frac{{\sigma}^{2}}{2}}\right)}^{2.382}$ | class 1 | 6.19 | 24.47 | 5.19 |

class 2 | 24.69 | 27.67 | 3.49 | |

class 3 | 45.28 | 18.17 | −0.62 |

**Table 4.**Allometric models to estimate the commercial volume (V

_{com}in m

^{3}) at tree level, considering a stump height of 15 cm. The explanatory variables are DBH (cm) and/or h (tree total height in meters).

Allometric Model | RMSE (m^{3}) | Relative RMSE (%) | Bias (%) |
---|---|---|---|

${V}_{com}={e}^{\left(-11.1974+3.2301.\mathrm{ln}\left(DBH\right)\right)}{e}^{\frac{{0.3164}^{2}}{2}}$ | 0.0574 | 25.74 | 11.27 |

${V}_{com}={e}^{\left(-12.0131+3.5689.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.3878}^{2}}{2}}$ | 0.0741 | 33.22 | 12.72 |

${\mathrm{V}}_{com}={e}^{\left(-12.8904+1.2739.\mathrm{ln}\left(DB{H}^{2}h\right)\right)}{e}^{\frac{{0.2746}^{2}}{2}}$ | 0.0482 | 21.60 | 8.64 |

* ${\mathrm{V}}_{com}={e}^{\left(-12.8904+1.2739.\mathrm{ln}\left(f{\left(h\right)}^{2}h\right)\right)}{e}^{\frac{{0.2746}^{2}}{2}}$ | 0.0608 | 27.24 | 1.43 |

**Table 5.**Allometric models to estimate tree stem volume (V

_{stem}in m

^{3}). The explanatory variables are DBH (cm) and/or h (m).

Allometric Model | RMSE (m^{3}) | Relative RMSE (%) | Bias (%) |
---|---|---|---|

${V}_{stem}={e}^{\left(-9.2146+2.6742.\mathrm{ln}\left(DBH\right)\right)}{e}^{\frac{{0.1796}^{2}}{2}}$ | 0.0413 | 21.25 | 1.03 |

${V}_{stem}={e}^{\left(-9.6807+2.8732.\mathrm{ln}\left(h\right)\right)}{e}^{\frac{{0.2271}^{2}}{2}}$ | 0.0503 | 25.88 | 2.54 |

${V}_{stem}={e}^{\left(-9.5344+0.9334.\mathrm{ln}\left(DB{H}^{2}h\right)\right)}{e}^{\frac{{0.1373}^{2}}{2}}$ | 0.0319 | 16.41 | 0.37 |

* ${\mathrm{V}}_{stem}={e}^{\left(-9.5344+0.9334.\mathrm{ln}\left(f{\left(h\right)}^{2}h\right)\right)}{e}^{\frac{{0.1373}^{2}}{2}}$ | 0.0461 | 23.72 | −1.11 |

**Table 6.**Average values of some dendrometric variables grouped by DBH classes. The standard deviation of the observations is noted in brackets. Sample size = 456 trees (see Table 1).

DBH Class | Tree Stem Commercial Volume (m^{3}) | Tree Stem Dry Biomass (kg/tree) | AGB (kg/tree) |
---|---|---|---|

5–10 cm | 20.52 (4.74) | 25.65 (5.93) | |

10–15 cm | 42.08 (18.57) | 52.60 (23.22) | |

15–20 cm | 0.1483 (0.056) | 125.98 (28.43) | 157.48 (35.54) |

20–25 cm | 0.3036 (0.075) | 205.46 (39.58) | 256.82 (49.47) |

25–30 cm | 0.4885 (0.061) | 302 (37.60) | 378.45 (47.01) |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aguilar, F.J.; Nemmaoui, A.; Peñalver, A.; Rivas, J.R.; Aguilar, M.A. Developing Allometric Equations for Teak Plantations Located in the Coastal Region of Ecuador from Terrestrial Laser Scanning Data. *Forests* **2019**, *10*, 1050.
https://doi.org/10.3390/f10121050

**AMA Style**

Aguilar FJ, Nemmaoui A, Peñalver A, Rivas JR, Aguilar MA. Developing Allometric Equations for Teak Plantations Located in the Coastal Region of Ecuador from Terrestrial Laser Scanning Data. *Forests*. 2019; 10(12):1050.
https://doi.org/10.3390/f10121050

**Chicago/Turabian Style**

Aguilar, Fernando J., Abderrahim Nemmaoui, Alberto Peñalver, José R. Rivas, and Manuel A. Aguilar. 2019. "Developing Allometric Equations for Teak Plantations Located in the Coastal Region of Ecuador from Terrestrial Laser Scanning Data" *Forests* 10, no. 12: 1050.
https://doi.org/10.3390/f10121050