Site Index Equations Research Articles

Density Management Diagram for Northeastern Red Spruce and Balsam Fir Forests

Abstract A stand-density management diagram is presented for use in northeastern red spruce and balsam fir forests. The diagram was derived from an extensive archived data set collected during the 1970s from fully stocked stands throughout northern Maine and a more recent study of precommercially thinned stands. The negative exponential relationship between mean stemwood volume per tree and stand density, commonly known as the "self-thinning rule, "was formulated to define a biological maximum stand density. The maximum size-density equation can be used to calculate the relative density of any stand and is accurate for thinned and unthinned natural stands as well as plantations. Equations for estimating quadratic mean diameter and stand top height are also derived for unthinned natural stands only. Data used to fit the self-thinning line are substantially above the A-lines on the familiar northeastern stocking guides, suggesting that these guides underestimate maximum density and thus overpredict self-thinning. Examples illustrate how to use the diagram to predict stand development under commercial and precommercial thinning scenarios, as well as natural stand development without thinning. Relevant site index and volume equations are included in an appendix. North. J. Appl. For. 16(1):48-56.

Construction of site index equations for Pinus sylvestris L. using permanent plot data in Sweden

Three general methods for construction of site index equations (the guide curve method, the parameter prediction method and the difference equation method) and 13 growth functions were tested on data from permanent plots in 156 Scots pine ( Pinus sylvestris) stands. Most stands were of cultivated origin with known year of establishment. Data consisted of measured top height and total age at the start and at the end of observation periods of different lengths. Functions and construction methods were compared with respect to residual variation (root mean square error (RMSE), meter) from original data. The guide curve method gave RMSE ∼ 0.56 while the difference equation method with the anamorphic formulation gave RMSE ∼ 0.46, with appropriate functions. The difference equation method gave a higher RMSE with polymorphic than with anamorphic formulation. The special formulation of the Hossfeld function for the difference equation method proposed by Cieszewski and Bella [Cieszewski, C.J., Bella, I.E., 1989. Polymorphic height and site index curves for lodgepole pine in Alberta. Can. J. Res. 19, 1151–1160] gave a simple and straigtforward, polymorphic solution with RMSE = 0.40. The parameter prediction method, in combination with the deviation method proposed by Tveite [Tveite, B., 1969. A method for construction of site-index curves. Meddelelser fra Det Norske skogforsoksvesen 27, 134–159] and a four-parameter growth model, gave the lowest RMSE, but this solution had patterns of overfitting. Parameter prediction with the Mitscherlich function gave an equation with almost the same accuracy as the polymorphic difference equation with the Hossfeld function. These functions deviate much at higher ages, but too few observations in old stands in primary data prevented their separation. Published data from yield plots in old, natural stands supported the Hossfeld function. But, indications of a changed growth pattern in later decades still leave open the question of height growth pattern in old pine stands of cultivated origin. The presented Hossfeld equation can only be recommended for use in 10–80-year-old pine stands of cultivated origin.

Measurement error causes bias in site index equations

Modeling height in the context of site index involves two processes: (i) estimating height at base age (site index), given height at some other age, and (ii) estimating height at some desired age given height at base age. When these two processes are modeled by individual functions for each process, height is assumed to be measured without error when on the right-hand side of the equation, but possessing error when on the left-hand side of the equation. This assumption causes a bias in the parameters of the site index curves. The bias in the estimated parameters is evident in the magnitude of the curves and in the shape of the curves, the typical change of shape of the curves from low site index to high site index is underestimated. An ad hoc method is presented that solves the problem. The ad hoc method is conceptually related to methods used in linear regression and to maximum likelihood estimation in a measurement errors problem context. A general measurement errors model is presented as an alternative. The measurement errors model requires an independent estimate of measurement error.

Development of a well-behaved site index equation: jack pine in north central Ontario

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Preliminary Site Index Equations for Three Planted Species in Northern Ontario

Abstract Stem analysis data from plantations of black spruce, white spruce, and jack pine from northern Ontario were fitted to base-age specific and base-age invariant site index models. The resulting equations and their respective parameters were compared via nonlinear analysis of covariance. The base-age specific models produced a somewhat better fit to the data than their base-age invariant counterparts, although the latter are considered theoretically more elegant. Graphical comparison of plantation and natural stand site index curves for the three species showed both similarities and differences. North. J. Appl. For. 12(2):57-63.

Comparison of the modified Weibull and Richards growth function for developing site index equations

A site index model based on the Weibull cumulative distribution function was fitted to three stem analysis data sets of black spruce (Picea mariana (Mill) BSP), white spruce (Picea glauca (Moench) Voss) and aspen (Populus tremuloides Michx.). The resulting equations were compared with those based on the modified Richards' (1959) biological growth function (Ek 1971; Payandeh 1977, 1978). The Weibull-type model performed as well as or better than the biological growth function for white spruce and aspen, respectively, but it performed less accurately in the case of black spruce. The Weibull-type model appears to behave better, i.e., it produces more realistic height growth patterns, than the Richards function upon extrapolation particularly at young ages.

A bi-segmental curve fitting approach to improve the accuracy of site index equations

Following Newnham's suggestion, Payandeh and Wang remondified Ek's base-age specific site index equation so as to satisfy the definition of site index, i.e. to meet the condition of H = S at index age. However, imposing such a constraint adversely affects the accuracy of the parameter estimation of the model, i.e. it results in a larger sum of squared error. A bi-segmental curve fitting procedure is proposed here to improve the accuracy of Payandeh and Wang's model when fitted to stem analysis data.

Application of the Kalman filter model in site index equation construction

Extension of the Richards biological growth function has been applied extensively to describe height growth patterns or to develop site index equations. We introduce a linear filter model to achieve the same goals. Although derivation of the filter model involves more computation, it compares well with the Richards growth model in accuracy when applied to a black spruce (Piceamariana (Mill.) B.S.P) stem analysis data set from northern Ontario. The main advantage of the filter model, however, is that it can be used as a base-age invariant site index model.

Modified site index equations for major Canadian timber species

Modified site index equations for major Canadian timber species were developed based on constrained biological and the Weibull-type growth functions. The resulting equations satisfy the definition of site index, height being exactly equal to site index at index age, with only slight loss of accuracy compared with the unconstrained model used previously. Direct estimate of site index from stand height and age is accomplished through numerical method rather than by a separate regression equation, as was done earlier.

A numerical method for the solution of a base-age-specific site index model

A numerical solution was derived for previously published site index equations for trembling aspen (Populustremuloides Michx.). Data from aspen stands in north central Ontario were used to verify the validity and accuracy of the numerical method. With the numerical solution, only the height-growth equation is employed to estimate (i) height from site index and age and (ii) site index from height and age. The numerical method simplified site index estimation and improved its accuracy for aspen stands in north central Ontario.

Cottonwood Site-Index Growth Curves

Cottonwood Site-Index Growth Curves

Stem Volume and Site Index Equations for European Larch in Maine

Abstract Stem analysis data collected from 101 sample trees located in 12 plantations established between 1930 and 1982 throughout central Maine were used to develop total and merchantable stem volume prediction equations, and site index prediction equations for plantation-grown European larch. The inside bark merchantable volume equation (4 in. top dob and 12 ft minimum merchantable bole) using a weighted combined variable was very similar to one for Japanese larch in Pennsylvania. Site index curves from this study were identical to those developed in southern New York and New England below a breast height (bh) age of 20 yr; after bh age 20, our curves predicted increasingly greater height growth and show a 6-12 ft superiority in height at a bh age of 50. North. J. Appl. For. 10(2):70-74.

Yield prediction models and comparative growth rates for six eucalypt species

Summary Stand stem volume yield prediction models are presented for six eucalypt species Eucalyptus regnans F. Muell., E. obliqua L'Her., E. delegatensis R. Baker. E. pilularis Smith, E. grandis Hill ex Maiden and E. diversicolor F. Muell. The data set used was collected over many years by eight of the major forestry agencies of Australia. It included data only from fully stocked, monospecific stands which had received no major natural or unnatural disturbance throughout their life-times. The bulk of the data was collected from regrowth forest, but some data from plantation forest were also included. The data did not sample formally or fully the entire area of natural occurrence of the six species. These data limitations must be borne in mind when applying the models. A site-index equation is developed for each species and these are then used in developing the stand stem volume yield prediction models which were based on the commonly used Schumacher model. The models are then used to compare growth rates of these species. It is shown that E. diversicolor has the most rapid early growth rate in stem volume and E. pilularis has the slowest. The other species are intermediate in growth rate between these two and do not differ greatly between each other.

Development of a well-behaved site index equation: jack pine in north central Ontario

A base-age invariant site index equation for jack pine based on the Chapman–Richards function was produced that satisfied nine criteria of preferred behavior for site index equations. A difference form of the Chapman–Richards equation produced the best behavior; height equalled site index at base age, and the shape of the curves reflected the data. The data structure used to fit the difference equation was all possible differences rather than the conventional nonoverlapping sequential intervals because this improved the behavior of the model. Height-prediction equations typically use height at base age (site index) as a predictor variable. As site index is measured with error, the equation will be biased. This bias will be evident in the predicted height at base age and in the shape of the curves. Base-age invariant equations predict height and site index with the same equation and thus diminish the effect of stochastic predictor variables. The equation performed comparably to a previously published equation with a specific base age of 50 years.

Dominant Height and Site Index Equations for Douglas-Fir and Engelmann Spruce in East-Central Arizona

Abstract Dominant height and site index equations were constructed for Douglas-fir (Pseudotsuga menziesii var. glauca) and Engelmann spruce (Picea engelmannii) on the Fort Apache Indian Reservation, located in east-central Arizona. An indirect parameter prediction method was utilized to develop the equations from stem analysis data. The dominant height equation for Douglas-fir is a function of site index, age, habitat type groups, and soil texture groups. The Engelmann spruce dominant height equation is a function of only site index and age. Site index may be calculated directly by inverting the dominant height equations. West. J. Appl. For. 7(2):40-44.

Base-Age Invariant Polymorphic Site Index Curves for Black Spruce and Balsam Fir Within Central Newfoundland

Abstract Increasing management intensity within Newfoundland requires that regional site index equations be species-specific and base-age invariant. Accordingly, base-age invariant polymorphic site index curves for black spruce and balsam fir were developed for use on upland mineral sites within central insular Newfoundland. Specifically, employing mean dominant height and breast-height age data derived from 129, 0.081 ha permanent sample plots situated in pure black spruce (n = 71) and mixed black spruce/balsam fir (n = 58) stands, site index equations were calibrated by species and stand type. In addition, given the approximate numeric equivalence of the resultant parameter estimates, data for both species and stand types were combined and a pooled equation calibrated. The equations were evaluated on their bias and predictive ability over 10-, 20-, and 30-year projection periods. Results indicated that the predictions were positively biased averaging +1.29 m, +2.36 m, and +3.15 m for 10-, 20-, and 30-year projection periods, respectively. Corresponding 95% prediction error intervals for these periods were -0.56 to +3.14 m, -0.88 to +5.44 m, and -1.27 to +6.56 m. These biases and prediction errors are within acceptable limits for site-class designations; however, explicit use of the site-index equations with growth and yield models would depend on error tolerances specified by individual users. North. J. Appl. For. 9(1):18-22

Tree and stand volume functions for Eucalyptus saligna

Tree and stand volume equations are developed for Eucalyptus saligna plantations in Western Kenya. Total tree volume, merchantable volume to a variable top diameter limit, and taper functions are developed using the logarithmic model. Separate equations are necessary for coppiced and non-coppiced plantations. A single equation is developed which predicts total yield per hectare and merchantable yield per hectare to any top diameter limit. A set of anamorphic site index equations and a stand basal area growth curve are developed. The system of equations indicates that mean annual increment culminates at approximately age five to six across the range of site indices present in these plantations.

National site-index and height-growth curves for white spruce growing in natural stands in Canada

Based on stem analyses, national site-index equations and height-growth equations for white spruce (Piceaglauca (Moench) Voss) were developed covering an area from the Yukon to Newfoundland. The objective was to construct a single set of curves for each relationship by combining all regions, instead of separate curves for individual provinces or regions. The results of these tests indicated that the objective could be reached without any significant loss of accuracy. In developing the equations a modified form of the Chapman–Richards function was employed, and results are presented in tabular and graphical forms.

Dominant-height and site-index equations for ponderosa pine in east-central Arizona

A dominant-height equation for ponderosa pine (Pinusponderosa Laws.) was constructed utilizing a parameter prediction method and a model by J.E. King. The site-index equation was obtained by inverting the dominant-height equation. A method was examined for indirectly obtaining parameter prediction equations. It proved superior to two direct parameter prediction approaches. Other site-quality variables, such as habitat type groups and elevation, were included in the dominant-height and site-index equations and were successful in refining predictive capability.

Polymorphic site index equations for four New Brunswick softwood species

As part of a provincial forest site classification program, polymorphic site index equations were developed for Pinusbanksiana Lamb., Piceaglauca (Moench) Voss, Piceamariana (Mill.) B.S.P., and Abiesbalsamea (L.) Mill. in New Brunswick using a generalized version of the Chapman–Richards' height-growth model and data from stem analysis of dominant and codominant trees growing on a wide range of sites. Two generalized five-parameter versions of the Chapman–Richards' model were tested. The resulting equations explain from 96 to 98% of the variation in dominant tree height using breast-height age and site index as independent variables.