Generalized Algebraic Difference Approach Research Articles

Dynamic Height Growth Equations and Site Index-Based Biomass Models for Young Native Species Afforestations in Spain

The expansion of forested areas through afforestation and reforestation is widely recognized as a highly effective natural solution for mitigating climate change. Accurately assessing the potential carbon uptake capacity of newly afforested areas requires modelling tools to estimate biomass stocks, including site index curves and biomass models. Given the unique conditions in terms of tree size, uniform spacing, and tree allometries observed in young afforestations compared to natural stands, specific tools are necessary. In Spain, over 800,000 ha has been afforested with native forest species since 1992, but specific modelling tools for these plantations are lacking. Using data from 370 stem analyses collected across an extensive network of plots in young afforestations, we developed dynamic height growth and site index models for the main native species (five pines and five oaks) commonly used in afforestation in Spain. We compared various nonlinear models, such as ADA (algebraic difference approach) and GADA (generalized algebraic difference approach) expansions. The developed site index models were then used to predict the total biomass stored in the afforestation. Our results underscore the necessity for specific site index models tailored to afforestations, as well as the potential of the established site index in predicting biomass and carbon fixation capacity in these young forests.

Dynamic height growth models for highly productive pedunculate oak (Quercus robur L.) stands: explicit mapping of site index classification in Serbia

Key messageWe applied the generalized algebraic difference approach (GADA) to develop dynamic models of height growth for pedunculate oak (Quercus robur L.) in Serbia. According to the dominant heights, the studied region comprises some of Europe’s most productive sites for pedunculate oak. Therein, we have generated a map showing the current site index class of stands. Such a map could be used to enhance forest management and evaluate climate change impacts.ContextAlthough sustainable forest management requires reliable prediction of forest site productivity, such indicators are currently unavailable for pedunculate oak sites in Serbia. The site index (SI) curves represent the most commonly used indirect scale for site productivity classification. The dynamic equations derived by the Generalized Algebraic Difference Approach (GADA) are the state-of-the-art approach in growth modeling, but they have not been widely applied for studying the height dynamics of pedunculate oak.AimsThe main objectives of this study were to develop the first dynamic site index curves for pedunculate oak in Serbia and subsequently to provide stand-level maps with predicted site indices.MethodsWe have tested five flexible polymorphic equations with variable asymptotes derived by the GADA approach. Models were calibrated using artificially established growth trajectories obtained from 3636 detailed temporary sample plots. The selection of the most suitable model was accomplished according to (1) quantitative measures of goodness of fit, (2) the analysis of residual scattering, and (3) the biological plausibility of obtained height growth curves.ResultsAfter correcting the error terms with a continuous first-order autoregressive structure and conducting a three-stage performance analysis, the GADA dynamic site index model derived from the Hossfeld base equation shows the best overall properties. Insight into the oscillations of relative error suggested that 100 years is the most suitable age for site index referencing. Comparison with existing height growth models revealed greater flexibility and a considerably better representation of the height growth dynamic of pedunculate oak in the studied region. Additionally, we have produced a spatially explicit map showing the expected SI100 for 1907 stands with pedunculate oak within 22 management units.ConclusionDynamic SI-curves based on GADA will serve forest practitioners to update management plans and serve as a reference point for benchmarking the impact of climate change and for developing adaptation strategies. The utilized approach allowed unbiased estimation of SI100 across all age classes so that the results could be mapped at a broader scale. This study provides the second known application of the dynamic model for pedunculate oak in Europe but the first that includes some of the most productive sites in the species distribution range.

Site Index Models for Loblolly Pine Forests in the Southern United States Developed with Forest Inventory and Analysis Data

Abstract Accurate productivity estimates are essential to assess the overall sustainability of forest resources. Site index (SI) models for loblolly pine (Pinus taeda L.) in plantation and natural forests of the southeastern United States were developed using the Forest Inventory and Analysis (FIA) database. We extracted short (~20 years), unbalanced panel data from the FIA database. Ten different nonlinear models derived from the base models using the algebraic difference approach (ADA) or the generalized algebraic difference approach (GADA) were fitted to the extracted data. The performance of the models was ranked based on a variety of fit and evaluation statistics. The results showed that all top three models were derived using the GADA approach. The best model for loblolly pine plantation and natural forest stands was derived from the Hossfeld model and the Chapman–Richards model, respectively. The best-fitted models for planted forests were also compared with previously developed models. This study demonstrated that base-age invariant and polymorphic SI models could be developed using short panel data extracted from FIA data. The SI models presented here can be used as a height growth model component in forest growth and yield model systems. Study Implications: Improved site index equations for assessing the site quality of loblolly pine plantation and natural stands are now available to stakeholders at the policy, management, and operational levels. Activities such as forest management, restoration, and wildlife management, which require site quality data, will benefit from the new models. Furthermore, the approach of deriving panel data based on Forest Inventory and Analysis data offers information on developing and updating models for other species. Finally, the approach of this study, to use permanent plot measurement data in developing growth and yield models, is cost-effective and time-efficient.

Laurel Regeneration Management by Smallholders to Generate Agroforestry Systems in the Ecuadorian Amazon Upper Basin: Growth and Yield Models

Laurel (Cordia alliodora Ruiz & Pav. Oken) is a Neotropical native tree that is easily regenerated in the secondary forest within the Amazon region. Amazonian smallholders use this tree regeneration to obtain a homogeneous forest cover when developing local agroforestry systems, which do not depend on nursery seedling production for tree planting. The objective of the present investigation was to develop growth and yield models for Laurel within the local agroforestry systems. A total of 226 sampling plots were measured between 2010–2011 and 2014–2015. Chapman- Richard, Hosslfeld II, and the generalized algebraic difference approach (GADA) form of the Chapman-Richard’s function was used for modeling height-age and diameter-age relationships. Eight volume models were tested to describe total stem volume. The GADA method was suited to describe the Laurel height and diameter-age growth. The cutting cycle for agroforestry systems with a density of 300 trees ha−1 at the best site index (SI) (22 m) produced 13.9 m3 ha−1 year−1 and a total wood yield of 195.1 m3 ha−1 at age 14. In the worst SI (14 m), the average annual yield was 3.5 m3 ha−1, with a total yield of 83.3 m3 ha−1 at age 24 years. The Spurr potential model was the best fit to describe the volume of the Laurel according to the Akaike information criteria. The Laurel biological (optimal) rotation age suggests that the minimum cutting diameter should be lowered from 30 cm of DBH in the research zone. Management of the natural regeneration of secondary forests by smallholders is a local agroforestry practice that should be given greater attention, especially within protected forest areas.

Site classification for Eucalyptus sp. in a tropical region of Brazil.

The aim was to determine the productive capacity of a forest site by applying different methods of fitting, combined with geostatistical techniques, to stands of Eucalyptus sp. in a tropical region of Brazil. Data were collected from 845 plots of a continuous forest inventory over four years. The classification of local production capacity was performed using growth curves obtained by the guide curve (GC) method, algebraic difference approach (ADA) and generalized algebraic difference approach (GADA) methods and ordinary kriging through the spherical, exponential, and Gaussian models to determine the spatial dependence of the variables site, geographical boundaries of site index classes, and their respective areas for each hectare in an annual production unit (APU). The modified Chapman-Richards model, fitted by the generalized difference approach method (GADA), provided the best statistical results, an improvement of 12.23% and 39.80% on the ADA and GC methods, respectively. The exponential model selected to express the spatial distribution of dominant height showed a high degree of spatial dependence.

Forecasting production in thinned clonal stands of Tectona grandis in Eastern Amazonia

Aim of the study: We investigated the most suitable thinning ages and intensities to maximize productivity and minimize the rotation age of Tectona grandis clonal plantations in the Brazilian Eastern Amazon.
 Area of study: Capitão Poço, State of Pará, Eastern Amazonia, Brazil.
 Materials and methods: We used diameter, height, and volume data from 72 permanent plots measured on nine occasions. We determined the classification of forest sites using the generalized algebraic difference approach (GADA). Clutter’s segmented model was used to simulate different intensities of basal area reduction, determining the technical ages according to the projected increments.
 Main results: The polymorphic site curves generated by the GADA method revealed that there were sites with different productive characteristics. The Clutter model produced compatible projections of basal area and volume that followed the behavior of the productivity classes. The final production was maximized when three thinning intensities (basal area reductions) were applied: 1st thinning (50%), between the ages of 3.5 to 4.2 years; 2nd thinning (50%), between the ages of 6.1 to 7.3 years; and 3rd thinning (25%), between the ages of 10.6 to 12.8 years. Projected rotation ages ranged from 13.9 to 16.6 years earlier than seminal plantings. The simulations increased the net volume by 8.3%, on average, compared to no thinning.
 Research highlights: Simulations with three thinnings maximized production compared to the no thinning scenario. The time interval between thinnings was less than five years. The research results can help forest management decision-making and reveal production increases of T. grandis clonal stands in a shorter time.

A Dynamical Model Based on the Chapman–Richards Growth Equation for Fitting Growth Curves for Four Pine Species in Northern Mexico

Tree growth models describe the growth and development of forest ecosystems by considering how the dimensions of each simulated tree change within a certain time. These models have commonly used three growth parameters that describe various biological processes and behaviours, considering a sigmoid growth function: (i) the upper asymptote (θ1), which is the maximal yield indicated by a final dimension (such as the maximal stem diameter); (ii) the maximum specific growth rate (θ2), defined as the slope of the tangent at the inflexion point; and (iii) the time elapsed (θ3), defined by the intercept of this tangent with the abscissas. To the best of our knowledge, however, associations between the three parameters have not been documented for tree species. Using diameter growth data from pine trees located in typical mixed and uneven-aged pine-oak forests in the Sierra Madre Occidental, Mexico, our study aims were: (i) to quantify the putative associations between the three growth parameters and (ii) to test the accuracy of a proposed Hybrid Chapman-Richards growth model based on associations between the three growth parameters, but including only one single parameter, relative to the widely used Generalized Algebraic Difference Approach (GADA) based on the Chapman-Richards, Lundqvist and Hossfeld models and the Hybrid Weibull Model. For statistical comparison of the quality of the models, we used the mean relative percentage error, root mean square error, coefficient of determination and Akaike information criterion to assess the quality of the fit. Although the quality of the five growth models studied was similar, from a practical point of view, the proposed Hybrid Chapman-Richards Model (CR-H) is easier to apply than the other models and has a lower data collection and computational cost. The parameter of CR-H can be easily obtained, by measuring just the dominant trees, especially in coniferous forests with irregular ages. Moreover, in contrast to the Chapman-Richards-GADA factor χ0, when θ2 is assumed to be site-specific, the CR-H has always a closed-form solution.

Nueva ecuación dinámica de altura dominante e índice de sitio para Pinus chiapensis (Martínez) Andresen en Puebla y Veracruz, México

Introduction: Pinus chiapensis (Martínez) Andresen lacks in many areas of site index (SI) models to classify timber productivity. Objective: To develop a dynamic SI equation, using the generalized algebraic difference approach (GADA) to describe the dominant height growth pattern and classify the productivity of natural stands of P. chiapensis in Puebla and Veracruz, Mexico. Materials and methods: Four theoretical growth models were used to derive six equations in GADA, fitted to dominant height-age observations from stem analyses of 31 trees. Fitting was performed using the Dummy variable method, which is invariant to the base age; autocorrelation and heteroscedasticity were corrected. Results and discussion: Quantitative evaluation, graphical analysis of residuals and growth trends of equations allowed the selection of an equation derived from the Levakovic II model with higher predictive capacity. With this equation and a base age of 50 years, polymorphic SI curves with variable asymptotes were constructed to classify productivity into low, medium and high, corresponding to SI of 25, 32 and 39 m, respectively. The maximum mean annual increase for the SI of 32 m was 1.07 m∙yr-1 and occurred at 11.08 years. The equation exhibited better performance relative to a previously reported polymorphic equation. Conclusions: It is recommended to use the developed equation to predict dominant height growth and SI of P. chiapensis stands in Puebla and Veracruz, Mexico.

Climatic and Topographic Variables Improve Estimation Accuracy of Patula Pine Forest Site Productivity in Southern Mexico

Sustainable forest management requires accurate biometric tools to estimate forest site quality. This is particularly relevant for prescribing adequate silvicultural treatments of forest management planning. The aim of this research was to incorporate topographic and climatic variables into dominant height growth models of patula pine stands to improve the estimation of forest stand productivity. Three generalized algebraic difference approach (GADA) models were fit to a dataset from 66 permanent sampling plots, with six re-measurements and 77 temporary inventory sampling plots established on forest stands of patula pine. The nested iterative approach was used to fit the GADA models, and goodness-of-fit statistics such as the root mean square error, Akaike’s Information Criterion, and Bias were used to assess their performance. A Hossfeld IV GADA equation type that includes altitude, slope percentage, mean annual precipitation, and mean annual minimum temperature produced the best fit and estimation. Forest site productivity was negatively affected by altitude, while increasing the mean annual minimum temperature suggested the fastest-growing rates for dominant tree height.

An Inverse Growth Curve Representation of the Clutter-Jones Stand Survival Model

Abstract The Clutter-Jones (CJ) stand survival model has been widely used for the projection of the reduction in trees per unit area in a forest stand over time. To better understand this well-recognized model, we introduced a varying-parameter formulation as an alternative to its original difference equation form, where plot-specific effects are presented explicitly. This new parameterization turns out to be a type of Richards (generalized logistic) equation with its independent variable age log-transformed, which might be termed “log-Richards” (LR) and falls into the category of the generalized algebraic difference approach (GADA) models. Under the LR GADA representation, a half-reduction age, at which half of the trees in a stand die, may be examined to improve stand survival simulation. The new reformulation provides an inverse growth curve modeling viewpoint of stand survival over time. It may motivate one to derive new survival models directly from existing growth equations that allow for generating inverse growth curves, complementary to the commonly used differential equation method in developing stand mortality models. Study Implications: The Clutter-Jones stand survival model plays an important role in southern pine plantation management as a component of whole-stand modeling systems. This study introduces a varying-parameter formulation as an alternative to its original difference equation form, providing an inverse growth curve modeling viewpoint of stand survival over time.

Thinning-response modifier term in growth models: An application on clonal Tectona grandis Linn F. stands at the amazonian region

Thinning is a widely used silvicultural method in the Tectona grandis Linn. F. (teak) stands, however, limited work has been done on modeling for predicting both short- and long-term responses to this treatment. Dynamic growth models for teak stands in Easter Amazon were developed and evaluated from data of 67 plots measured between two and ten years old. We tested the models of Chapman-Richards, Hossfeld, and Lundqvist-Korf using the generalized algebraic difference approach (GADA) to describe the basal area of the stand, including the effect of thinning, which was quantified from a silvicultural treatment response function. Dominant height was described using a dynamic equation derived onwards by Lundqvist-Korf model, considering one parameter to be site specific, while the stand volume was described as of Chapman-Richards model using the theory of covariates. We used 80% of the data for fitting and 20% for validating the developed models. The following fitting statistics were used: Akaike's information criterion, square root of the mean percent error and estimation efficiency. For the validation, was used the equivalence test (regression based TOST using bootstrap). The equation derived from the Chapman-Richards function from GADA integrated with a thinning modifier provided the best performance in describing stand dynamics. The developed thinning modifier were shown to significantly improve predictions of stand-level basal area growth, and the results showed that models that disregard silvicultural treatment cannot be used to obtain reliable projections of the stands under analysis. The growth models provided detailed and accurate description of the dynamics of teak stands in the Amazon region, allowing for a well-defined maximum volume rotation. Overall, the study highlights the importance of including thinning modifiers in growth and yield models, especially for shade intolerant species, such as teak.

Two subject specific modelling approaches to construct base-age invariant polymorphic site index curves with varying asymptotes

For constructing growth and yield models the concept of site index as measure of productivity is crucial. Here, we use nonlinear mixed-effects models (NLME) with random individual effects and nonlinear models with dummy variables as fixed individual effects (NLFE) to fit mechanistic growth functions to stem analysis data of the economically most important tree species in Zhongtiaoshan forest region, China. The Richards and Lundqvist function are formulated into five dynamic equations (R1, R2, L1, L2 and L3) applying the generalized algebraic difference approach (GADA), which inherit polymorphism, varying asymptotes and base-age invariance. According to Akaike information criterion the R1 model as NLFE fits height growth data of Carrière, Franch., Koidz., Blume and Sukaczev best, while for Blume R2 as NLFE fits height growth data best. For Mayr L1 as NLME has been selected as best model, as R1 and R2 both as NLFE and NLME are not extrapolating the comparably short length of height growth data well enough. However, according to the root mean square error and bias differences between model fits of both the selected equation and the chosen model fitting approach are not so clear. Presented families of height growth curves serve as planning tools to identify site index and therefore assess productivity of forest stands in the studied region. A direct comparison of the productivity of forest stands of the same tree species is possible due to base-age invariance of the selected models.Pinus tabuliformisPinus armandiiQuercus liaotungensisQuercus alienaBetula platyphyllaQuercus variabilisLarix principis-rupprechtii

Modeling dominant height growth using permanent plot data for Pinus brutia stands in the Eastern Mediterranean region

Aim of the study: At current, forest management in the Eastern Mediterranean region is largely based on experience rather than on management plans. To support the development of such plans, this study develops and compares site index equations for pure even-aged Pinus brutia stands in Syria using base-age invariant techniques that realistically describe dominant height growth.Materials and methods: Data on top height and stand age were obtained in 2008 and 2016 from 80 permanent plots capturing the whole range of variation in site conditions, stand age and stand density. Both the Algebraic Difference Approach (ADA) and the Generalized Algebraic Difference Approach (GADA) were used to fit eight generalized algebraic difference equations in order to identify the one which describes the data best. For this, 61 permanent plots were used for model calibration and 19 plots for validation.Main results: According to both biological plausibility and model accuracy, the so-called Sloboda equation based on the GADA approach showed the best performance.Research highlights: The study provides a solid classification and comparison of Pinus brutia stands growing in the Eastern Mediterranean region and can thus be used to support sustainable forest management planning.Keywords: site index; Generalized Algebraic Difference Approach (GADA); Sloboda equation.

Developing a site index model for P. Pinaster stands in NW Spain by combining bi-temporal ALS data and environmental data

Site index (SI) is a common measure of forest site productivity, serving as a valuable baseline for forest management. The main objective of this study was to develop a SI model for Pinus pinaster Ait. in north-west Spain by combining bi–temporal, low–density airborne laser scanning (ALS) data (acquired in the periods 2009–2011 and 2015–2017) with climatic, edaphic and physiographical data. Site productivity, assessed by site quality curves, was modelled using an age-independent difference equation method based on ALS metrics and environmental variables. For the model development process, we used data from 156 sample plots in pure and even-aged P. pinaster stands distributed throughout Galicia (NW Spain) and measured in the Spanish National Forest Inventory (SNFI). The generalized algebraic difference approach (GADA) formulation was tested by using two different base equations for modelling the dominant height growth (ΔH) from ALS variables. The GADA formulation derived from the Bertalanffy’s base model produced the best estimates of dominant height (H) for P. pinaster stands in Galicia. Use of the proposed model to estimate ΔH for a new pine stand requires two ALS data sets for estimating site-specific (local) parameters. To enable use of the model when such information is not available, the relationship between the values of the site-specific parameter and environmental variables was described using Multivariate Adaptive Regression Splines (MARS). Use of the MARS equation enabled us to develop spatially-explicit predictive maps of the site-specific parameter values, which can be used together with the GADA model to derive ΔH curves and SI estimates for P.pinaster stands in the whole study region.

Dynamic top height models for several major forest tree species in Great Britain

Abstract In this paper, we present new dynamic top height models for 10 major species in Great Britain. These models for pure, even-aged stands replace the current, static top height-age curves published in the 1980s, which are no longer reliable in the light of the new data that have been added over the past decades. We followed the generalized algebraic difference approach (GADA) to build the new models and specifically focused on models derived from the Hessfeld IV and Chapman–Richards height-age base equations. We selected a multi-asymptote and polymorphic GADA version of the Chapman–Richards equation for all the species, with the anamorphic functional form as a special case derived from the non-significance of one of the global parameters. The new models are dynamic, accurate and robust, which will permit unbiased yield predictions from current top height-dependent yield simulators in Great Britain.

Un modelo aproximado de crecimiento en altura e índice de sitio para Quercus sideroxyla Bonpl. en rodales mezclados de Durango, México

Introduction: Predictions of dominant or co-dominant height are an important element in planning forest management with timber production objectives. Objective: To develop dominant height growth and site index (SI) equations for Quercus sideroxyla Bonpl. Materials and methods: The height current annual increment model was fitted, and the age of the sections was estimated from a database of 29 stem-mapped plots. A taper analysis data of 37 trees was also used to reconstruct dominant height growth trajectories. Three equations based on algebraic difference approach (ADA; one anamorphic and two polymorphic) and an equation based on generalized algebraic difference approach (GADA) were used to simultaneously model the dominant height and SI. Results and discussion: The ADA polymorphic equations were statistically better than the anamorphic equation, according to the adjusted coefficient of determination, root mean square error, mean bias, Akaike’s information criterion and log-likelihood; however, the GADA equation was better than the ADA equations. The growth curves were biologically realistic and showed SI classes at a base age of 60 years. Conclusions: The developed equations can be used for decision making in forest management when Q. sideroxyla shows dominance over Pinus species and the main objective is timber production.

Assessing site form as an indicator of site quality in even-aged Pinus radiata D. Don stands in north-western Spain

Key messageSite form and site index have shown similar precision for estimating site quality in even-agedPinus radiataD. Don stands in north-western Spain. Additionally, SF presents the advantage that it does not require stand age information and can therefore be used in a wider set of situations in the forestry practice.ContextEstimation of site quality is essential for characterizing, monitoring and predicting forest resources. Site index (i.e. the dominant height of the stand at a reference age) is ordinarily used to estimate site quality; however, this index is only useful for even-aged stands of known age. By contrast, SF is age-independent as it uses the dominant height of the stand at a reference dominant diameter.AimsThe aim of this study was to compare the performance of SF and SI for site quality estimation in even-aged P. radiata stands.MethodsDynamic equations derived with the Generalized Algebraic Difference Approach (GADA) from the Hossfeld IV base model were fitted to predict site quality with both SI and SF. SF predictions were compared with SI regarding variability within the same plot and consistency in site quality predictions, using the observed maximum mean annual volume increment (MAI) as a direct measure of site quality.ResultsBoth approaches showed good performance in model fitting and provided similar goodness-of-fit statistics and variability in the predictions. However, SI performed slightly stronger when related to MAI.ConclusionSF performed adequately in estimating site quality for even-aged P. radiata stands, with results comparable to those obtained using traditional SI.

Growth and Yield Models for Balsa Wood Plantations in the Coastal Lowlands of Ecuador

Balsa trees are native to neotropical forests and frequently grow on fallow, degraded land. Balsa can be used for economic and ecological rehabilitation of farmland with the aim of restoring native forest ecosystems. Although Ecuador is the world’s largest producer of balsa, there is a lack of knowledge about production indicators for management of balsa stands in the country. The aim of this study was to develop growth and yield models (i.e., site index (SI) curves and stem volume models) for balsa plantations in the coastal lowlands of Ecuador. Balsa trees growing in 2161 plots in seven provinces were sampled. Here we present the first growth and yield models for the native, although underutilized, balsa tree. Three curve models were fitted to determine SI for balsa stands, differentiating five site quality classes. Eight volume models were compared to identify the best fit model for balsa stands. The mean annual increment was used to assess balsa production. The generalized algebraic difference approach (GADA) equation yielded one of the best results for the height–age and diameter–age models. The Newnham model was the best volume model for balsa in this comparative study. The maximum annual increment (i.e., for the best stand index) was reached in the second year of plantation. The fitted models can be used to support management decisions regarding balsa plantations. However, the models are preliminary and must be validated with independent samples. Nevertheless, the very fast development of the native balsa tree is particularly promising and should attract more attention from forest owners and politicians.

Modeling Dominant Height Growth of Cedar (Cedrus libani A. Rich) Stands in Turkey

Abstract Sustainable forest management requires accurate prediction from a growth and yield system. Such a system relies heavily on some measure of site productivity, which is often the site index. A model was developed for predicting dominant height growth and site index of even-aged cedar (Cedrus libani A. Rich.) stands in Turkey. Stem-analysis data from 148 trees were used for model development and validation. Six dynamic height–age equations were derived using the generalized algebraic difference approach (GADA). Autocorrelation was modeled by expanding the error term as an autoregressive process. Based on numerical and graphical analysis, a GADA formulation derived from the Chapman–Richards model was selected. Based on relative error in dominant height prediction, 80 years was selected as the best reference age. The resulting equation provided the best compromise between biological and statistical aspects and, therefore, is recommended for height growth prediction and site classification of cedar stands in Turkey.

SITE QUALITY ESTIMATIONS BASED ON THE GENERALIZED ALGEBRAIC DIFFERENCE APPROACH: A CASE STUDY IN ÇANKIRI FORESTS

ABSTRACT In this study, it is aimed that the dynamic site index models were developed for Crimean Pine stands in Sarikaya-Cankiri forests located in middle northern Turkey. The data for this study are 153 sample trees obtained from the Crimean Pine stands. In modeling relationships between height and age of dominant or co-dominant trees, some dynamic site index equations such as Chapman-Richards (M1, M2, M3), Lundqvist (M4 and M6), Hossfeld (M5), Weibull (M7) and Schumacher (M8) based on the Generalized Algebraic Difference Approach (GADA) were used. The estimations for these eight-dynamic site index model parameters with well as various statistical values were obtained using the nonlinear regression technique. Among these equations, the Chapman-Richards’s equation, M3, was determined to be the most successful model, with accounted for 89.03 % of the total variance in height-age relationships with MSE: 1.7633, RMSE: 1.3279, SSE: 1165.6, Bias: -0.0380. After determination of the best predictive model, ARMA (1, 1) autoregressive prediction technique was used to account autocorrelation problems for time-series height measurements. When ARMA autoregressive prediction technique was applied to the Chapman-Richards function for solving autocorrelation problem, these success statistics were improved as SSE: 868.7, MSE: 1.3183, RMSE: 1.1482, Bias: -0.06369, R2: 0.918. Also, Durbin-Watson statistics displayed that autocorrelation problem was solved by the use of ARMA autoregressive prediction technique; DW test value=1.99, DW<P=0.5622, DW>P=0.4378. The dynamic site index model that was developed has provided results compatible with the growth characteristics expected in the modeling of height-age relations, such as polymorphism, multiple asymptote, and base-age invariance.