Volume Ratio Equations Research Articles

Stem volume ratio equations to variable merchantable limits for sweet chestnut in Portugal

Stem volume ratio equations to variable merchantable limits for sweet chestnut in Portugal

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

AbstractTen 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.

Generalized stem taper and tree volume equations applied to eucalyptus of varying genetics in Brazil

Lack of generalized equations has prevailed in Brazil, because it is assumed that localized or climate-specific equations are needed. This study aimed to develop generalized stem taper and volume equations applicable to 11 eucalyptus clones and evaluate if climate variation impacts the accuracy of the estimates. A total of 693 trees evenly distributed across 11 clones at 21 sites were used for model fittings and predictive validation. The penalized mixed spline (PMS) approach was developed for predicting stem taper and volume along the stem profile. The Schumacher and Hall (1933) equation was used to predict total tree volume, while volume ratio equations were applied to predict merchantable volume. For every fitted equation, an annual climatic variable was included to assess the improvement in model performance. The overall results highlighted that climatic variation does not need to be accounted for in stem taper and volume modeling. All of the equations displayed desirable accuracy, but the generalized PMS equation may be preferred when the forestry enterprise looks to furnish a range of multiple forest products. The generalized total tree volume equation, combined with the ratio equations, is highly recommended when the forestry enterprise produces a single product.

Taper and Volume Systems Based on Ratio Equations for Pinus pseudostrobus Lindl. in Mexico

Studies on functional relationships between relative stem diameter and height to estimate timber yield are useful in the management of commercial forest plantations. With taper analysis data of 42 Pinus pseudostrobus Lindl. trees in the indigenous community of Nuevo San Juan Parangaricutiro , Michoacan, Mexico, six compatible systems for predicting taper (d), merchantable volume (Vm), stem volume (Vs), total tree volume (Vt) and branch volume (Vb) were fitted and evaluated. The compatible taper and merchantable volume equations were based on volume ratio equations. Three Vs equations were tested in each system. In general, the compatible systems presented statistical accuracy in the d, Vm, Vs and Vt components but were less accurate in Vb. Three compatible systems were selected, according to their more efficient goodness-of-fit statistics, and a different total volume equation was incorporated into each system. The compatible systems based on volume ratio equations are simple, reliable tools for estimation of stand timber stocks and product classification of P. pseudostrobus in commercial forest plantations in Mexico.

Deriving compatible taper functions from volume ratio equations based on upper-stem height

A very common model for prediction of tree stem volumes to upper-stem height or diameter limits is the use of a merchantable to total volume ratio function multiplied by a total stem volume function. Many users of these prediction systems also desire taper equations that can predict heights to upper-stem diameters. While taper equations compatible with volume ratio equations have been used for many years, compatible taper equations from volume ratio equations that are functions of upper-stem height have been used infrequently. Yet many studies have indicated that height-based ratio equations perform well and frequently have statistics of fit that are comparable with diameter-based volume ratio equations. Compatible taper equations derived from height-based ratio equations are presented here. The methodology that uses height-based merchantable to total volume ratios does not require the solution of a differential equation after differentiating the height-based volume ratio, as is necessary when using the method of deriving taper equations from diameter-based merchantable to total volume ratios. This could be an advantage depending on the complexity of the ratio function. Example taper equations fitted to loblolly pine (Pinus taeda L.) data from the southeastern USA and the state of Oklahoma, USA, indicate good fit to these data, whether fitted directly to taper data or implicitly by using parameters fitted to volume ratio data.

Improved accuracy of aboveground biomass and carbon estimates for live trees in forests of the eastern United States

Accurate estimation of forest biomass and carbon stocks at regional to national scales is a key requirement in determining terrestrial carbon sources and sinks on United States (US) forest lands. To that end, comprehensive assessment and testing of alternative volume and biomass models were conducted for individual tree models employed in the component ratio method (CRM) currently used in the US' National Greenhouse Gas Inventory. The CRM applies species-specific stem volume equations along with specific gravity conversions and component expansion factors to ensure consistency between predicted stem volumes and weights, and additivity of predicted live tree component weights to match aboveground biomass (AGB). Data from over 76 600 stem volumes and 6600 AGB observations were compiled from individual studies conducted in the past 115 years – what we refer to as legacy data – to perform the assessment. Scenarios formulated to incrementally replace constituent equations in the CRM with models fitted to legacy data were tested using cross-validation methods, and estimates of AGB were scaled using forest inventory data to compare across 33 states in the eastern US. Modifications all indicated that the CRM in its present formulation underestimates AGB in eastern forests, with the range of underestimation ranging from 6.2 to 17 per cent. Cross-validation results indicated the greatest reductions in estimation bias and root-mean squared error could be achieved by scenarios that replaced stem volume, sapling AGB, and component ratio equations in the CRM. A change in the definitions used in apportioning biomass to aboveground components was also shown to increase prediction accuracy. Adopting modifications tested here would increase AGB estimates for the eastern US by 15 per cent, accounting for 1.5 Pg of C currently unaccounted for in live tree aboveground forest C stock assessments. Expansion of the legacy data set currently underway should be useful for further testing, such as whether similar gains in accuracy can be achieved in estimates of regional or national-scale C sequestration rates.

Variable-top merchantable volume equations for Scots pine (Pinus sylvestris) and Sitka spruce (Picea sitchensis (Bong.) Carr.) in Northern Britain

Summary In this study, data collected from stands in Scotland and Northern England were used to develop variable-top merchantable volume equations for plantation-grown Scots pine (Pinus sylvestris) and Sitka spruce (Picea sitchensis). After fitting and evaluating a number of total volume and volume ratio equations, the Schumacher and Hall (1933) equation (Schumacher, F.X. and Hall, F.S. 1933 Logarithmic expression of timber tree volume. J. Agric. Res. 47, 719–734) gave the best results for total volume, while the Van Deusen et al. (1981) and Cao and Burkhart (1980) models (Van Deusen, P.C., Sullivan, A.D. and Matney, T.G. 1981 A pre diction system for cubic foot volume of loblolly pine applicable through much of its range. South. J. Appl. For. 5, 186–189; Cao, Q.V. and Burkhart, H.E. 1980 Cubic-foot volume of loblolly pine to any height limit. South. J. Appl. For. 4, 166–168) gave the best fits for volume ratio equations to variable-top diameter and height limits, respectively. Variable-top merchantable volume equations were jointly estimated as a product of total volume and volume ratio equations. The final models, obtained using a non-linear mixed-effects modelling approach, showed considerable improvements compared with their fixed-effects counterparts. Even though further modelling of serial correlation using a banded toeplitz error structure improved model fits, it also led to an increase in the residual variance of the models and could degenerate the predictive ability of the models. We therefore recommend the use of mixed models with simple ‘iid’ error structure. However, in situations where additional data are not available to predict the random effects, we recommend the use of the models based on non-linear least squares.

THEORETICAL BASIS FOR SLURRY COMPUTATION AND COMPOUNDING IN HIGHLY DEVIATED WELLS CEMENTING

Highly deviated oil and gas wellbores offer higher recovery and productivities than vertical wells. But achieving cement slurry stability and accurate volume in vertical wells is not as critical as in deviated wells chiefly because of the large overburden pressure resting on the slurry in deviated wells, among other reasons. Good slurry performance would require the use of appropriately calculated slurry volume and cement materials and additives that can optimize simultaneously the placement and mechanical properties of set cement. In most cases today, slurry volumes in deviated wells are based on rule-of-thumb and utilization of local enlargement factors. In this case, successful job is judged based on slurry returns obtained at the surface. Furthermore, slurry strength is compromised due to excessive overburden and massive casing weight acting at an angle. It is therefore necessary to search for a more appropriate slurry recipe to mitigate slurry failure due to excessive overburden pressure, casing weight and inadequate slurry volume. This paper presents a to derivation of accurate area and volume ratio equations based on well deviation to calculate appropriate slurry volumes and suggests an idealized slurry systems in deviated sections of the well based on point masses to explore an appropriate slurry recipe to achieve high slurry integrity. Results show that angle of deviation is very critical to correct cement slurry volume computation for deviated wells. There is a non-linear relationship between cross-sectional area, volume ratio and angle of deviation. As the angle tends to 90 0 , the volume of slurry required increases compared to an equivalent volume required for an equivalent true vertical depth of a vertical well. Furthermore, particle point mass analysis shows that unequal cement particle size distribution would help to create slurry stability through reduction in interstitial water required thus sustain slurry integrity.

Modelling the diameter distribution of eucalyptus plantations with Johnson’s S B probability density function: parameters recovery from a compatible system of equations to predict stand variables

Introduction The simulation of diameter distributions is the basis for predicting volume in the so-called diameter distribution models. Combined with volume, volume ratio and taper equations, these models allow the prediction of volume assortments according to user needs. The simulation of diameter distributions is also essential in initialising individual tree models. It is also a useful aid for planning harvesting operations. Methods In this paper, Johnson’s distribution was used to model the diameter distribution of Eucalyptus globulus in Portugal. When a predefined probability density function is used as part of a growth and yield model, the parameters of the function must be estimated for each year during the simulation period. Results The development of a system of equations that relates stand characteristics to mathematical functions of the distribution, such as the moments of the distribution, allows for the estimation of parameters (i.e. parameter recovery). This method assures compatibility between the characteristics of the observed population used in parameter recovery and those obtained through simulation. Conclusions The system of equations was built in such a way that the observed well-established biological processes between stand variables is maintained, and the equations were simultaneously fitted to minimise the determinant of the covariance matrix of errors. Based on validation with an independent data set, the model provides precise estimates of total stand volume.

A system for compatible prediction of total and merchantable volumes allowing for different definitions of tree volume

A system of equations for compatible prediction of total and merchantable volumes that allows for different definitions of tree volume was developed in this study. The use of the developed system will allow the conversion and subsequent comparison of results from forest inventories using different definitions of tree volume (e.g., including or not the top material of the tree and (or) the stump, inside or outside bark). The compatibility between taper, total volume, and volume ratio equations is ensured by properly integrating the taper equation. The diameter under the bark at any height is modeled with the Demaerschalk taper equation, and the corresponding diameter over the bark is obtained by assuming that bark thickness is also modeled with Demaerschalk’s function. The set of equations that has contemporaneous cross-equation error correlation (known as nonlinear seemingly unrelated regression equations) was fit using nonlinear joint generalized least squares regression. The predictive ability was evaluated using an independent data set. The system is consistent and performs well when applied to maritime pine ( Pinus pinaster Ait.) trees in Portugal, showing better performance than do other total volume equations for maritime pine used in the latest Portuguese national forest inventories.

Volume ratio models for prediction of merchantable volumes of Radiata pine in the Basque Country

Four volume ratio equations were fitted as part of a Pinus radiata yield tables updating project in plantations within the Basque Country. In this approach, volume ratio equation gives the proportion of merchantable volume to total volume using final diameter and breast height diameter as independent variables. For models fitting, data from 71 felled trees were used. Trees were cubed by the Smalian formula, obtaining from each one both, the total volume over bark and the merchantable volume over bark for 7, 10, 15, 20, 25 and 30 cm of final diameter. Despite the four models exhibited a high precision to estimate merchantable individual volume, the model 1 (Parresol) had the best performance. They are considered original for this species in Spain.

A ratio method for predicting stem merchantable volume and associated taper equations for Cupressus lusitanica, Ethiopia

Volume ratio method is one of the methods used for estimating merchantable volume of a stem between a pre-specified stump height and top diameter. Volume ratios of merchantable volume to total volume were expressed as a function of the ratio of merchantable top diameter to breast height diameter ( d m/ D) and as a function of merchantable height to total tree height ( h m/ H). A number of mathematical equations were developed using d m/ D and h m/ H as independent variables for its estimation for Cupressus lusitanica. Fit index and standard error of estimate were used to evaluate and chose the best equations for each explanatory variable. The predictive abilities of the selected equations for estimating merchantable volume ratios were then examined using a test data. The results showed that the selected equations had an overall mean bias of −0.0111 and −0.0052, respectively. Development of merchantable volume ratio equations led to stem taper equations that predict diameter on a stem at any height and height at any diameter, based on given D and H. The performances of the derived taper equations were also tested. The overall absolute biases were 0.2276 cm and 0.2438 m for predicting merchantable top diameter and height at various relative heights and varying merchantable top diameters, respectively.

A volume prediction system for Quercus robur L. in Galicia (Northwestern Spain) based on a volume ratio equation

Eight total-volume equations and four volume-ratio equations (for diameter limits) were fitted to a sample of 225 trees of Quercus robur of different ages. The trees were collected in stands growing in a great variety of site qualities and densities in Galicia (Northwest of Spain). The data employed to develop volume ratio equations have a longitudinal structure so a second order autoregressive model was used to correct the error dependence implicit in this kind of data. Several «goodness-of-fit» statistics were used in order to select the best equations within each group (total volume equations and volume ratio equations) and cross-validation was used to analyse the performance of the model. Finally, the models selected were combined to develop a volume equation with product classification.

Compatible Volume and Taper Models with Coefficients for Tropical Species on Hainan Island in Southern China

Abstract Compatibility of a volume-taper equation system is generally defined to mean that volumes determined by integrating the taper curve from groundline to total tree height are algebraically identical to those obtained from the total volume or appropriate volume ratio equations. This constraint was applied to a previously published model as a boundary condition using the constant of integration resulting from the solution of the differential equation. Four cases are discussed that apply to different solutions of the differential equation. This set of models, all based on the same geometric assumption, makes a very flexible system for modeling stem taper. We fitted the resulting taper models and corresponding tip volume equations simultaneously to stem-measurement data for 23 tropical species growing on Hainan Island in southern China. The simplest case among the four (case I-a) was acceptable for some species, but for some others, a more general case (case II-b) was needed. We compared the results for the simplest and the most general cases of this system with a published geometric model having only a point constraint at dbh and with a published variable exponent taper equation. We examined how well each one predicted upper-stem diameters, merchantable height for a given upper-stem diameter, and total stem volume. Our taper models always had the smallest bias and standard error of estimates for total volume estimation. The variable exponent taper equation was the best for predicting upper-stem diameter and merchantable height. The advantages of a taper system with a geometric rationale are discussed, and two examples of easy extensions of our taper model are shown in the appendix. For. Sci. 45(1):85-100.

Individual tree merchantable volume to total volume ratios based on geometric solids

Individual tree volume ratio equations were developed on the basis of the volume formulas of certain geometric solids. Two parameter-free and two parameterized models were formulated, based on different basal diameters. The parameter-free models should be useful when few individual tree volume ratio data exist for a given species or for a particular population of trees. The parameterized models are appropriate, of course, to those situations where volume ratio data exist. The parameter-free models performed as well as the parameterized models in the upper half of the tree stem for the loblolly pine data used for comparison. The parameterized models performed well all along the tree stem.

A Ratio Method for Calculating Stem Volume to Variable Merchantable Limits, and Associated Taper Equations

One method for estimating the volume of the merchantable portion of a tree stem is the volume ratio method. In this study the ratio used with this method was derived from another ratio expressed as ground-to-limit volume to total stem volume. Because of the latter ratio's strong correlation with the ratio of merchantable top diameter/breast height diameter and merchantable height/total tree height, several mathematical models were formed employing these variables for its estimation for red pine (Pinus resinosa Ait.) and sugar maple (Acer saccharum Marsh.). Among the models that were tested, the best for each variable was chosen by assessing its adequacy by the statistical measures of fit index and standard error of estimate. When checked against an independent set of data, these best models performed satisfactorily, with small overall biases. Furthermore, taper equations were derived from these ground-to-limit volume ratio equations.

Stand-Level Cubic-Foot Volume Ratio Equations for Planted Loblolly Pine on Site-Prepared Land in the Mid-South Gulf Coastal Plain

Abstract Volume ratio predictor equations are presented for calculating cubic foot volumes per acre to any top diameter limit in loblolly pine plantations on cutover site-prepared land. An illustrative example of the application of the equations is presented. South. J. Appl. For. 12(1):7-11.

A volume and taper prediction system for bald cypress

A volume and taper prediction system based on d10 and consisting of a total volume equation, two volume ratio equations (one for diameter limits, the other for height limits), and a taper equation was developed for bald cypress using sample tree data collected in Louisiana. Normal diameter (dn), a subjective variable-height measure applied to bald cypress (Taxodiumdistichum (L.) Rich.) in place of diameter at breast height (dbh), was found to be inferior to five different fixed-height diameter measurement points in terms of predicting total volume. Diameter measured at 10 ft (3.0 m) above the ground, termed d10 is recommended as a better diameter measurement point for bald cypress. A number of "goodness-of-fit" statistics were employed to evaluate alternative functions for predicting volume and taper. Two statistics, bias and sum of squared relative residuals, provided the best discrimination between functions.

Tree Profile and Volume Ratio Equations for Loblolly Pine Trees on Cutover Site-Prepared Lands

Abstract Equations are presented for predicting the height and cubic foot volume (ib or ob) to any top diameter limit (ib or ob) of loblolly pine (Pinus taeda L.) trees grown on cutover site-prepared lands. Example applications of the equations are given. South. J. Appl. For. 10:241-244, Nov. 1986.

Tree Profile and Volume Ratio Equations for Sweetgum and Cherrybark Oak Trees

Abstract Equations are presented for predicting the height and cubic foot volume (ib or ob) to any top diameter limit (ib or ob) of sweetgum and cherrybark oak trees. Example applications of the equations are given.