Height-diameter-age equation systems for Pinus arizonica Engelmann and Pinus durangensis Martinez in mixed-species stands in Durango, Mexico

Abstract Introduction: Total height (H) and diameter at breast height (DBH) are important variables in forest inventory and they are the basis for growth and yield systems. Objective: To generate three prediction and projection equation systems for Pinus arizonica Engelmann (Pa) and Pinus durangensis Martinez (Pd) in mixed-species stands in Durango, Mexico. Materials and methods: The outside-bark DBH equations as functions of the inside-bark DBH were developed and the H-DBH relationship was extended to three relationships with the use of age (A): H-DBH, H-A and DBH-A. The equation systems of H-DBH-A were developed from a database of 46 and 66 stem analysis trees with 601 and 760 longitudinal measurements of Pa and Pd, respectively. The equations were fitted with seemingly unrelated regression and Dummy variables approach with common and specific parameters. Results and discussion: The relationships showed significant accuracy in the assessed fitting statistics (adjusted coefficient of determination, root mean square error, Akaike’s information criterion, standard error of the estimate and bias). The inverse equations of the three relationships formed a global system of prediction and projection equations. Conclusions: The equations are useful for predicting and projecting H and DBH and they can be used as input variables in growth and yield models.

An approximate height growth and site index model for Quercus sideroxyla Bonpl. in mixed-species stands of Durango, Mexico

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

Compatible Taper and Stem Volume Equations for Pine Species in Mixed-Species Forests

Ten systems of compatible taper and outside-bark volume equations derived from upper-height-based volume ratio equations were compared with a used segmented-stem system of compatible taper and volume equations. All the systems were simultaneously fitted to cumulative volume data and taper data for Arizona pine, Aztec pine, Durango pine, Mexican white pine, and Smooth-leaved pine species in naturally regenerated mixed-species forests in Mexico. The systems derived from volume ratio equations performed better than the segmented-stem system for most species. The resultant top systems might be easier to implement than the segmented-stem system for predicting upper-stem height, upper-stem diameter, merchantable volume, and total stem volume for considered species.

Can an Exponential Function Be Applied to the Asymptotic Density–Size Relationship? Two New Stand-Density Indices in Mixed-Species Forests

Abstract: This study presents two stand-density indices (SDIs) based on exponential density decline as a function of quadratic mean diameter for all species combined in mixed-species forests with 22 species mix grouped in four species groups. The exponential-based density–diameter relationship, as well the density index corresponding to the slope or instantaneous mortality rate parameters, was compared with those based on power-law density–diameter relationship. A dataset of 202 fully stocked circular plots at maximum density was used for fitting the models, and a dataset of 122 circular plots was used for validation stand density index for all species combined of mixed-species stands. The dataset for validation was independent of dataset for model development. The first stand-density index showed a density management graphic (DMG) with a variable intercept and common instantaneous mortality rate, and the second index showed a DMG with common intercept and variable mortality rate. Additionally, the value of the initial density of the fitted line was more realistic than those generated by the potential model for all species combined. Moreover, the density management diagrams showed a curvilinear trend based on the maximum stand density index in graphical log–log scale. The DMGs could be interpreted as forest scenarios based on variable initial density and common management objectives or the same density and different management objectives for forest-rotation periods involving all species combined in mixed-species stands. The fitting of exponential and potential equations for species or species groups showed that the density–size relationships in mixed-species forests should be modeled for all species combined because the disaggregation of mixture species represented a weak tendency for each species or species group and the resultant fitted equations were unrealistic.

Compatible taper, volume, green weight, biomass and carbon concentration system for Quercus sideroxyla Bonpl./Sistema compatible de ahusamiento, volumen, peso verde, biomasa y concentración de carbono para Quercus sideroxyla Bonpl.

Introduction: Estimation of total and merchantable tree volume, as well as of biomass and carbon, implies the generation of biometric tools essential in forest management and planning.
Objectives: To fit a compatible taper, volume, green weight, dry biomass and carbon concentration system for Quercus sideroxyla Bonpl. species using wood density.
Materials and methods: A database of 522 diameter-height measurements, obtained from 37 trees, was used in the fitting equations. The compatible system (CS) was integrated by 34 equations, which were simultaneously fitted by generalized nonlinear least squares. Taper and volume were the base variables for estimating green weight, dry biomass and carbon concentration.
Results and discussion: All equations were compatible with the stem volume equation, and the merchantable equations with the taper and merchantable volume equations. The fit statistics showed the efficiency of the equations in global terms and by relative height classes.
Conclusions: The CS has the property of estimating taper, merchantable volume, green weight, dry biomass and carbon concentration at upper-height and by components (stem, total tree and branches).
Introducción: La estimación de volumen total y comercial de árboles, así como la de biomasa y carbono, implica la generación de herramientas biométricas esenciales en el manejo y planeación forestal.
Objetivos: Ajustar un sistema compatible (SC) de ahusamiento, volumen, peso verde, biomasa seca y concentración de carbono para la especie Quercus sideroxyla Bonpl., con el uso de la densidad de la madera.
Materiales y métodos: Una base de datos de 522 pares de diámetro-altura, obtenida de 37 árboles, se utilizó en el ajuste. El SC se conformó de 34 ecuaciones ajustadas simultáneamente por mínimos cuadrados generalizados no lineales. El ahusamiento y volumen fueron las variables base para la estimación del peso verde, biomasa seca y concentración de carbono.
Resultados y discusión: Todas las ecuaciones fueron compatibles con la ecuación de volumen de fuste, y las ecuaciones comerciales, con los parámetros del ahusamiento y volumen comercial. Los estadísticos de ajuste mostraron la eficiencia de las ecuaciones en términos globales y por clases de altura relativa.
Conclusiones: El SC posee la cualidad de estimar el ahusamiento, volumen comercial, peso verde, biomasa seca y concentración de carbono a una altura comercial y por componentes (fuste, total árbol y ramas).

How to correct the heteroscedasticity and autocorrelation of residuals in taper and height growth models?



In modeling of taper functions and dominant height growth with time series data, the presence of heteroscedasticity and autocorrelation in residuals is common. Variance Functions (varFunc) and correlation structures (corStruct) were used to correct heteroscedasticity and autocorrelation; both were combined and evaluated through taper and height growth equations for Pinus teocote in Durango, Mexico. A dataset of 51 stems analysis with 768 taper observations and 634 height growth observations was used. The varFuncs applied were: 1) power function (varPower); 2) exponential function (varExp); 3) constant plus power function (varConstPower); and 4) a combination of power and exponential functions (varComb). The corStructs were: compound symmetry (corCompSymm), autoregressive of order 1 (corAR1), continuous-time autoregressive of order 1 (corCAR1), autoregressive-moving average (corARMA2-0), corARMA1-1, corARMA2-1, corARMA2-2, corARMA3-1 and corARMA3-2. To fit the equations, the generalized nonlinear least squares method was used and evaluated with a rating system through: RMSE, R^2, AIC, BIC, LogLik, VC and average bias. According to the rating system, the best combinations for taper and height growth equations were 1-9, 2-5, 3-8 and 4-6 and 1-6, 2-9, 3-7 and 4-4,respectively. In the taper equation, only the combination 2-5 was homoscedastic with independent residuals, and the selected height growth equations were homoscedastic with independent residuals; the varFunc and corStruct had influence on the trajectories of site index curves.