Stand-level diameter distribution yield model for black spruce plantations

Forest Ecology and Management 209 (2005) 181–192 www.elsevier.com/locate/foreco

Stand-level diameter distribution yield model for black spruce plantations P.F. Newton a,*, Y. Lei b,1, S.Y. Zhang c,2 a

Analytical Stand Dynamics Research, Canadian Forest Service-Ontario, Natural Resources Canada, 1219 Queen St. East, Sault Ste. Marie, Ont., Canada P6A 2E5 b Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, People’s Republic of China c Resource Assessment and Utilization Group, Forintek Canada Corporation, 319 Rue Franquet, Sainte-Foy, Que., Canada G1P 4R4 Received 22 October 2004; received in revised form 21 January 2005; accepted 22 January 2005

Abstract The objectives of this study were to develop and demonstrate a stand-level diameter distribution yield model and associated algorithm for black spruce (Picea mariana (Mill.) B.S.P) plantations. Employing a parameter prediction approach within the context of a stand density management diagram (SDMD), model development consisted of four sequential steps: (1) obtaining maximum likelihood estimates for the location, scale and shape parameters of the Weibull probability density function (PDF) for 296 empirical diameter frequency distributions; (2) developing and evaluating parameter prediction equations in which the parameter estimates of the Weibull PDF were expressed as functions of stand-level variables employing stepwise regression and seemingly unrelated regression techniques; (3) explicitly incorporating the parameter prediction equations into the SDMD modelling framework; and (4) developing an associated PC-based algorithm and demonstrating its utility in density management decision-making. The results indicated that the parameter prediction equations described 74.4, 87.1 and 66.8% of the variation in location, scale and shape parameter estimates, respectively. Incorporating the parameter prediction equations into the structure of the SDMD enabled the prediction of the temporal dynamics of the diameter frequency distribution by density management regime, site quality and region. An algorithmic version of the model is provided as a decision-support aid in which forest managers are able to simultaneously contrast multiple density management regimes in terms productivity, product value and optimal site occupancy. # 2005 Elsevier B.V. All rights reserved. Keywords: Structural yield prediction; Three-parameter Weibull PDF; Seemingly unrelated regression; Productivity; Product value; Optimal site occupancy

* Corresponding author. Tel.: +1 705 541 5615; fax: +1 705 541 5701. E-mail addresses: [email protected] (P.F. Newton), [email protected] (Y. Lei), [email protected] (S.Y. Zhang). 1 Fax: +86 10 62888315. 2 Tel.: +1 418 659 2647; fax: +1 418 659 2922. 0378-1127/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2005.01.020

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1. Introduction Density regulation is one of the principal silvicultural tools utilized by forest managers within the Canadian Boreal Forest region (CCFM, 2002). Conceptually, density regulation or stand density management, is the process of controlling resource competition through density manipulation in order to realize specified management objectives. Operationally, density manipulation consists of regulating the number and arrangement of individual trees on a given forest site through initial spacing and (or) a temporal sequence of thinning events. Stand-level consequences to density management include (1) acceleration of stand operability status (i.e. minimizing the time to harvestable stand status as defined by threshold indices of usable fibre per unit area and individual tree size (Erdle, 2000)), (2) spatial and structural uniformity of the residual stand thus facilitating subsequent thinning operations (Weetman, 1997), and (3) increased individual tree size resulting in improved product values (Zhang et al., 1998). Stand density management is a complex process given the multitude of treatment options available to the silviculturist. Consequently, the development of dynamic stand density management diagrams (SDMDs) and their algorithmic analogues (e.g. Burk and Mack, 2002; Newton, 2003), has greatly facilitated stand density management decision-making. Basically, the SDMD is a decision-support tool that is used to determine the density management regime required for the realization of specified management objectives: for example, deriving density regimes in order to (1) attain specified volumetric yields at rotation age (Newton and Weetman, 1993, 1994), (2) provide wildlife habitat (Sturtevant et al., 1996), or (3) control ericaceous vegetation (Smith, 1989). Structurally, SDMDs consist of a number of functional and empirical quantitative relationships, which collectively represent the cumulative effect of various underlying competition processes on tree and stand yield parameters (e.g. reciprocal equations of the competition-density and yield-density effect; self-thinning rule; among others (Ando, 1962; Drew and Flewelling, 1977, 1979; Jack and Long, 1996; Newton, 1997). The temporal dependency of these processes is governed by the intensity of competition and site quality as expressed by relative density index and site index, respectively.

Although SDMDs have been developed for numerous commercially-important species throughout the world, their utility has been largely limited to evaluating density management outcomes in terms of mean tree size and stand-level volumetric yields (Newton, 1997). Recently, however, the maximizing product value in addition to volumetric yield has become an important management objective as exemplified by corporate trends in Sweden (e.g. Hagner, 2000) and North America (e.g. Brunsdon, 2000; Terry, 2000). However, product value calculations require an estimate of the underlying diameter distribution given the inherent relationship between monetary value and tree size (e.g. Zhang and Chauret, 2001). Recently, based on the SDMD modelling approach, Newton et al. (2004) developed a stand-level diameter distribution yield model for black spruce (Picea mariana (Mill.) B.S.P) stands employing the parameter prediction method (Hyink and Moser, 1983). Although the approach resulted in an acceptable stand density management decisionsupport aid, its utility was limited to unmanaged (natural) stand types situated within eastern Canada. Consequently, the objectives of this study were to develop and demonstrate a similar but enhanced decision-support model and associated algorithm, specifically for black spruce plantations. The resultant model enables the simultaneous contrasting of complex density management regimes in terms of productivity, product value, and optimal site occupancy.

2. Method Generally, the method consisted of four sequential steps: (1) obtaining maximum likelihood estimates (MLEs) for the location, scale and shape parameters of the Weibull probability density function (PDF) for empirical diameter frequency distributions derived from tree list data obtained from sample plots located in black spruce plantations; (2) developing parameter prediction equations in which the parameter estimates of the Weibull PDF were expressed as functions of stand-level variables employing stepwise regression and seemingly unrelated regression (SUR) techniques; (3) explicitly incorporating the parameter prediction equations into the SDMD modelling framework; and

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(4) developing an associated PC-based algorithm and demonstrating its utility in operational planning. 2.1. Data description The data base consisted of 296 diameter frequency distributions obtained from (1) 111 0.040 ha permanent sample plots (PSPs) located in plantations situated on upland sites throughout north-eastern and north-western Ontario (denoted the BeckwithRoebbelen PSP dataset), (2) four variable-sized (0.098–0.114 ha) experimental PSPs (EPSPs) located in an initial spacing trial situated on an upland site in north-western Ontario (denoted the Stanley Spacing Trial (vide Zhang et al., 2002)), and (3) similar to (2), four variable-sized (0.083–0.095 ha) EPSPs located in a matching spacing trial situated on an upland site in the same region (denoted the Thunder Bay Spacing Trial (vide Zhang and Chauret, 2001)). The PSPs data consisted of 269 measurements obtained in the following years: 1981, 1982 or 1983 (PSP establishment), 1988 (first remeasurement), and 1999, 2000 or 2001 (second remeasurement). The EPSPs data consisted of the following: (1) 12 measurements obtained in 1985 (EPSP establishment), 1993 (first remeasurement) and 1998 (second remeasurement), from four EPSPs established within initial spacing treatments of 1.8 m
1.8 m (n = 3) and 2.7 m
2.7 m (n = 9) at the Stanley Spacing Trial, and (2) 15 measurements obtained in 1983 (EPSP establishment), 1988 (first remeasurement), 1993 (second

Total age (A; year) was determined from the plantation establishment records. For each plot measurement, computations included the following: (1) calculating total biotic density (Nb; stems/ha); (2) calculating the total height for each tree employing a second-degree polynomial regression function derived from the D–H subset measurements; (3) given (2), calculating the mean dominant height (Hd; m) as defined as the mean height of the trees within the largest height quintile; (4) calculating individual tree volumes, mean volume ð¯v; dm3 Þ, total volume (Vt; m3/ha) and merchantable volume (Vm; m3/ha) using standard volume equations (Honer et al., 1983; n, merchantability limits consisted of a 0.1524 m stump-height and a 7.62 cm top diameter (inside-bark) for all stems greater than 9.5 cm in diameter at breast height); (5) calculating quadratic mean diameter (Dq; cm), basal area (G; m2/ ha) and relative density index (Pr; % (Newton and Weetman, 1993)); and (6) calculating site index (SI; m) based on the Hd and A measurements (i.e. SI = Hd at an A of 50 years (Payandeh and Wang, 1995)). 2.2. Data analysis The diameter distribution at each plot measurement was described by the three-parameter Weibull PDF (Eq. (1); Weibull, 1951) given its historical record in successfully describing unimodal diameter distributions within forest stands (e.g. Bailey and Dell, 1973; ´ lvarez Schreuder and Swank, 1974; Little, 1983; A et al., 2002).

8
 

< c D  a c1 Da c exp  f ðD; a; b; cÞ ¼ b b b : 0 remeasurement) and 1998 (third remeasurment), from four EPSPs established within initial spacing treatments of 1.8 m
1.8 m (n = 4) and 2.7 m
2.7 m (n = 11) at the Thunder Bay Trial. The data collected at the time of each plot measurement included the following: (1) diameter frequency distribution tally for all biotic trees greater than 1.5 cm in diameter at breast-height (D; 0.1 cm) by species; and (2) total height (H; 0.31 m) on a subset of approximately 10 black spruce trees per plot.

183

if a  D  /

(1)

if D < a

where a is the location parameter which is equivalent to the minimum diameter of the distribution, b is the scale parameter which is related to the range of the distribution, and c is the shape parameter which is related to the degree of skewness of the distribution. The corresponding three-parameter Weibull cumulative distribution function (CDF) is given by Eq. (2). 

Da c FðD; a; b; cÞ ¼ 1  exp  (2) b

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ˆ and c (ˆc) were Parameter estimates for a (ˆa), b (b) obtained via MLE employing SAS’s (1999) nonlinear programming procedure. Note, in cases were the initial estimation resulted in negative values for the location parameter, a was arbitrarily set to the minimum observed diameter and parameters b and c re-estimated. The parameter prediction method (Hyink and Moser, 1983) was used to estimate the parameters of the Weibull PDF from stand-level variables. Specifically, based on preliminary graphical and correlation analyses, parameter estimates aˆ , bˆ and cˆ were expressed as functions of Hd, Nb, Pr, Dq, G, v¯ , Vt (or their logarithmic equivalents), and each other. Stepwise regression analysis was used to identify a set of candidate functional forms based on a 0.01 risk level (Kilkki et al., 1989). Parameter estimates for the resultant system of equations were estimated simultaneously using SUR techniques given that SUR takes into account error covariances across the three equations and is asymptotically efficient in the absence of specification error (Zellner, 1962).

meter prediction equations for predicting b, c and a are given by Eqs. (3)–(5), respectively.

3. Results and discussion

3.1. Modelling approach

Table 1 summarizes the mensurational characteristics of the sampled stands including the parameter estimates for the Weibull PDF. The resultant para-

Analytically, three main methods have been used to develop stand-level diameter distribution yield models: (1) parameter prediction method in which the

bˆ ¼ 5:7489 þ 0:1175H d þ 0:5213Dq  0:6262 ln N b þ 0:0407Pr

(3)

cˆ ¼ 1:6953 þ 0:3363bˆ  0:0972H d  0:0606Dq þ 0:013V t  0:0402Pr

(4)

aˆ ¼ 0:7228  0:2125bˆ þ 0:0626Dq þ 0:0309¯v þ 0:0116G

(5)

The approximate multiple coefficients of determination and corresponding residual mean squared errors were respectively: 0.871 and 1.347 for Eq. (3); 0.668 and 0.641 for Eq. (4); and 0.747 and 0.629 for Eq. (5). The system weighted R2 for the SUR estimation was 0.791. Incorporating the parameter prediction equations into the SDMD modelling framework, as schematically illustrated in Fig. 1 (vide Table 2 for associated variable definitions and computations), resulted in a stand-level diameter distribution yield model (sensu Clutter et al., 1983).

Table 1 Summary of the mensurational characteristics of the stand-level variables including the parameter estimates for the Weibull PDF derived from the 296 plot measurements Parametera (units)

Mean

Standard deviation

Standard error

Minimum

Maximum

A (year) SI (m) Hd (m) Nb (stems/ha) Pr (%) Dq (cm) G (m2/ha) v¯ (dm3) Vt (m3/ha) Vm (m3/ha) aˆ bˆ cˆ

26.02 12.73 8.09 3526 37.58 8.39 18.95 32.54 82.16 58.43 0.87 7.55 2.52

– – – 1702 22.91 – 12.70 – 74.75 68.78 1.24 3.72 1.11

10.08 2.69 3.84 – – 4.29 – 42.81 – – – – –

8 6.30 1.59 475 32.46 0.76 0.06 0.04 0.05 0 0.01 0.52 0.28

48 20.54 17.32 10650 93.45 24.80 49.33 226.26 322.24 289.76 6.78 16.75 8.44

a

Denotations defined in the text.

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Fig. 1. Schematic illustration of the computational sequence used to recovery diameter frequency distributions within the context of the SDMD modelling framework. Refer to Table 2 for variable definitions and associated computational details.

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Table 2 Variable definitions and associated computational details associated with the schematic illustrated in Fig. 1 Variable Description and computational details RC SI

NI T2AðiÞ T2NðiÞ T3AðiÞ T3NðiÞ Hˆ dðtÞ

ˆ qðtÞ D vˆ¯ ðtÞ Pˆ r Nˆ bðtÞ Nˆ aðtÞ ˆ ðtÞ G Vˆ tðtÞ Vˆ mðtÞ bˆ ðtÞ cˆ ðtÞ aˆ ðtÞ Nˆ bðt; jÞ

Regional code: Newfoundland (1); New Brunswick (2); Quebec (3); and Ontario (4) Regional-specific site index (m; mean dominant height at a fixed base-age): Newfoundland: Eq. (3) in Newton (1992; 50 year breast-height age); New Brunswick: Eq. (9) in Ker and Bowling (1991; 50 year breast-height age); Quebec: Eq. (2.1.1.1) in Pre´ gent et al. (1996; 25 year total age); and Ontario: Eq. (1A) in Payandeh and Wang (1995; 50 year total age) User-specified initial planting density (stems/ha) Thinning regime 2: age (year) at the time of the ith thinning event Thinning regime 2: density reduction (stems/ha) during the ith thinning event Thinning regime 3: age (year) at the time of the ith thinning event Thinning regime 3: density reduction (stems/ha) during the ith thinning event Predicted mean dominant height at time t (m). Derived from the SI function employing age at time t (A(t)) values of 1, . . ., 50 year and the associated vector of parameter estimates, bSI (Eq. (3) in Newton, 1992; Eq. (9) in Ker and Bowling, 1991; Eq. (2.1.1.1) in Pre´ gent et al., 1996; or Eq. (1A) in Payandeh and Wang, 1995) Predicted quadratic mean diameter at breast height at time t (cm). Estimated employing an empirically-derived regression relationship where bDq denotes the parameter estimate vector (Eq. (3) in Newton and Weetman, 1994) Predicted mean volume at time t (dm3). Estimated employing an empirically-derived regression relationship where bv¯ denotes the parameter estimate vector (Eq. (4) in Newton and Weetman, 1994) Predicted relative density index at time t (%). Derived from the asymptotic size-density (self-thinning) relationship for black spruce where bPr denotes the parameter estimate vector (Eq. (2) in Newton and Weetman, 1994) Predicted biotic density at time t (stems/ha). Explicitly modelled using NI and Na(t) (Eq. (7) in Newton and Weetman, 1994) Predicted abiotic density at time t (stems/ha). Derived from an empirically-based mortality rate function (RM(t)) which is used to predict the number of trees (stems/ha) that will incur mortality per metre of height growth at time t where bM denotes the parameter estimate vector for the mortality rate function (Eq. (6) in Newton and Weetman, 1994) ˆ ðtÞ ¼ 0:00007854D ˆ 2 Nˆ bðtÞ Þ Predicted basal area at time t (m2/ha) ðG qðtÞ

Predicted total volume at time t (m3/ha) ðVˆ tðtÞ ¼ 1000vˆ¯ ðtÞ Nˆ bðtÞ Þ Predicted merchantable volume at time t (m3/ha). Estimated employing an empirically-derived regression relationship where bVm denotes the parameter estimate vector (Eq. (5) in Newton and Weetman, 1994) Predicted scale parameter of the Weibull PDF (Eq. (1)) at time t (Eq. (3) within text) Predicted shape parameter of the Weibull PDF (Eq. (1)) at time t (Eq. (4) within text) Predicted location parameter of the Weibull PDF (Eq. (1)) at time t (Eq. (5) within text) Predicted number of trees (stems/ha) within the ith diameter class (D(j) where j = 2, 4, . . ., 38) at time t (Eq. (2) in combination with Eqs. (3)–(5))

parameters of a PDF are estimated from stand-level variables (e.g. Clutter and Bennett, 1965; Smalley and Bailey, 1974a, 1974b; Dell et al., 1979; Feduccia et al., 1979; Schreuder et al., 1979; Little, 1983; Clutter et al., 1984; Rennolls et al., 1985; Kilkki and Pa¨ ivinen, 1986; Kilkki et al., 1989; Maltamo et al., 1995; Maltamo, 1997; Siipilehto, 1999; Leduc et al., 2001; Newton et al., 2004; This study); (2) parameter recovery method in which the parameters of the PDF are derived from the moments of the diameter distribution which are themselves estimated from stand-level variables (e.g. Matney and Sullivan, 1982; Knoebel et al., 1986; Lindsay et al., 1996); and (3) percentile approach in which the parameters of the PDF are derived from specified percentiles of the

diameter distribution which are themselves predicted from stand-level variables (e.g. Lohrey and Bailey, 1976; Magnussen, 1986; McTague and Bailey, 1987; Bailey et al., 1989; Brooks et al., 1992; Knowe, 1992). However, results from a recent analysis in which all three methods were compared, employing part of the same data set utilized in this study, revealed little difference among the methods in terms of prediction errors and goodness-of-fit statistics (Liu et al., 2004). Consequently, the employment of the parameter prediction approach was considered an acceptable approach in regards to developing a stand-level diameter distribution yield model for black spruce plantations. Furthermore, in terms of modelling resolution, the stand-level diameter distribution yield

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model represents the highest level possible for black spruce plantations given the non-spatially-explicit nature of the tree-level measurements currently available. The Weibull PDF has been frequently used to quantify the diameter frequency distribution within various forest types due to (1) its ability to describe a wide range of unimodal distributions including reversed-J shaped, exponential, and normal frequency distributions, (2) the relative simplicity of parameter estimation, and (3) its closed cumulative density functional form (e.g. Bailey and Dell, 1973; Schreuder et al., 1979; Little, 1983; Rennolls et al., 1985; Mabvurira et al., 2002; Newton et al., 2004). The results of this study provide continued support for use of the Weibull PDF in describing diameter structures, particularly within monospecific even-aged boreal conifer stand types. The use of SUR procedures resulted in more efficient parameter estimates than those that would be obtained via an equation-byequation ordinary least squares procedure given the explicit incorporation of contemporaneous error correlation which existed across the three equations (Zellner, 1962). 3.2. Algorithmic version and utility In order to facilitate the application of stand-level diameter distribution yield models in management planning, requires the development of computer software given the complexity of the computations involved. Consequently, an algorithmic version of the model was developed. Specifically, based on the relationships illustrated within Fig. 1 and described in Table 2, a PC-based program was developed using Lahey FORTRAN1 (F77L-EM/32, Version 5.20, Lahey Computer Systems Corporation, USA) and a number of graphical subroutines developed by Heart2PI hPJ PV ¼ 100
4

i¼1

187

land Software (HGRAPH, Version 5.00, Heartland Software Corporation, USA). Denoted SDMDMID2 (Stand Density Management Diagram for Managed stands – Interactive version – Diameter distribution recovery model), the program predicts, tabulates and graphically illustrates stand-level yield attributes and associated diameter frequency distributions (number of trees (stems/ha) per diameter class (2, 4, . . ., 38 cm)) for each of the following density management regimes: (1) planting without thinning (control); (2) planting with 1–4 thinning entries; and (3) planting with a second-set of 1–4 thinning entries. Additionally, the SDMDMID enables the user to simultaneously contrast the density management regimes in terms of three stand-level performance indicators: (1) overall productivity (merchantable mean annual increment (MAI; m3/ha/year)); (2) relative product value; and (3) degree of optimal site occupancy. Computationally, rotational MAI included the merchantable volumes extracted during the thinning events (Eq. (6)). P V mR þ Ii¼1 V mT ðiÞ MAI ¼ (6) AR where V mT ðiÞ is the merchantable volume (m3/ha) removed during the ith thinning entry (i = 1, . . ., I; I = 4), V mR is the standing merchantable volume (m3/ ha) at rotation, and AR is the stand age (year) at rotation. Product value is based on the cubic relationship between diameter and monetary value applicable to boreal conifers (Zhang et al., 1998; Zhang and Chauret, 2001): product value = f (diameter3) when diameter 9.6 cm. Stand-level product value was approximated by the additive sum of treatmentspecific and rotational values. The cumulative values for the thinning regimes are compared to the control regime in order to calculate a relative product value index (PV; Eq. (7)).

 i P  3 J 3 =ATðiÞ þ j¼10 N TR ðjÞ DTR ðjÞ =AR 5  100 P  J 3 j¼10 N CR ðjÞ DC ðjÞ =AR

3 j¼10 N TðijÞ DTðijÞ

(7)

R

2

1

Complies with the FORmula TRANslation (FORTRAN) 77 standard (ANSI, 1978).

The executable version of the program and required subroutines can be extracted from the SDMDMID.ZIP file available at ftp:// ftp.glfc.forestry.ca/pnewton/outgoing/.

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where N TðijÞ is the number (stems/ha) of trees removed during the ith thinning entry within the jth diameter class ( j = 10, 12, . . ., J; J = 38), DTðijÞ is the mid-point diameter (cm) of the trees removed during the ith thinning entry within the jth diameter class, ATðiÞ is the stand age (year) at the time of the ith thinning entry, N TR ðjÞ is the number (stems/ha) of trees within the jth diameter class left standing at rotation in the thinned plantation, DTR ðjÞ is the mid-point diameter (cm) of the trees within the jth diameter class left standing at rotation in the thinned plantation, N CR ðjÞ is the number (stems/ha) within the jth diameter class left standing at rotation in the unthinned plantation (control), and DCR ðjÞ is the mid-point diameter (cm) of

the trees within the jth diameter class left standing at rotation in the unthinned plantation. The degree of optimal site occupancy (SO; %) was based on the relative number of years in which the size-density trajectory was within an optimal Pr management zone (Eq. (8)): 0.3  Pr < 0.5 where Pr is relative density index (Newton, 2005).


NO SO ¼ 100
NM

 (8)

where NO is the number of years in which the sizedensity trajectory was within the optimal Pr management zone (0.3  Pr < 0.5), and NM is the total num-

Fig. 2. SDMD for black spruce plantations with thinning regimes superimposed. Principal component include: (1) approximate crown closure line (CC) at a relative density (Pr) of 0.13; (2) approximate lower limit of the zone of imminent competition-mortality (ZICM) at Pr of 0.50; (3) self-thinning rule at Pr of 1.0; (4) isolines for mean dominant height (HT), quadratic mean diameter (DBHOB) and merchantability ratio (MV/ TV); and (5) 50 year size-density trajectories for three user-specified density management regimes. SDMDMID.EXE program input: (1) regional code = 4 for Ontario; (2) site index = 18 m at 50 years total age; (3) regime 1: initial planting density = 3000 stems/ha; (4) regime 2: first thinning of 750 stems/ha @ 20 years and a second thinning of 750 stems/ha @ 40 years; and (5) regime 3: 1 thinning of 1500 stems/ha @ 30 years.

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ber of years since the size-density trajectory achieved a minimum Pr of 0.3. The SDMDMID provides forest managers with a tool for deriving yields and evaluating complex density management regimes. Specifically, the algorithm can assist a manager in selecting the optimal regime in terms of maximizing productivity, relative product value, or optimal site occupancy. For example, consider the following three density management regimes for plantations situated on good quality sites (18 m @ 50 years total age) in Ontario: regime 1: initial planting density = 3000 stems/ha; regime 2: initial planting density = 3000 stems/ha with 2 thinnings (750 stems/ha @ 20 years; and 750 stems/ha @ 40 years); and regime 3: initial planting density = 3000 stems/ha with 1 thinning (1500 stems/ha @ 30 years). The algorithm graphically illustrates the (1) temporal mean volume-density trajectories within the context of the underlying SDMD (e.g. Fig. 2), and (2) the corresponding recovered diameter frequency distributions (e.g. Fig. 3) from which MAI, PV and SO are calculated (e.g. Table 3). In this example, regimes 1–3 would be the most applicable in terms of attaining maximum productivity, product value, and optimal site occupancy, respectively. 3.3. Structural yield prediction Development of structural yield prediction models is a basic prerequisite for estimating multiple-product yields and associated monetary values, as historically shown for intensely managed southern pine species (e.g. Clutter and Bennett, 1965). Compared to average stand-level models, using the entire diameter distribution to estimate size-dependent attributes such as

189

Fig. 3. Graphical illustration of the corresponding diameter frequency distributions at ages 10 (cross), 20 (square), 30 (circle), 40 (triangle) and 50 (diamond) by regime where DBH refers to diameter at breast-height, and R1, R2 and R3 refer to regimes 1–3, respectively (st/ha = stems/ha).

product value is inherently more precise then using mean diameter and density alone. In addition, structural yield prediction estimates are useful in solving a board array of forest management problems. For example, knowledge of the diameter distribution

Table 3 Comparison of three density management regimes for plantations situated on good quality sites (18 m @ 50 years total age) in Ontario in terms of productivity, product value and optimal site occupancy criteria as determined from the SDMDMID algorithm Regime No.

Initial planting density (stems/ha)

No. of thinning treatments

Time of treatment (age)

Density reduction (stems/ha)

1 2

3000 3000

0 2

3

3000

1

0 20 40 30

0 750 750 1500

a b c

As defined by Eq. (6). As defined by Eq. (7). As defined by Eq. (8).

MAI (m3/ha/year)a

PV (%)b

SO (%)c

6.08

0.00

21.62

5.74 5.65

9.99 3.34

51.35 59.46

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at various stages of stand development is of utility in (1) designing thinning prescriptions (e.g. determining the number and size of trees to be removed and retained in order to realize specified residual stocking goals), (2) evaluating wildlife habitat potential (e.g. determining the potential number of avian cavity trees based on minimum diameter thresholds), (3) estimating carbon storage potential (e.g. stratifying rotational yields into forest products and determining their respective decay rates), and (4) assessing the attainment of biological diversity objectives (e.g. determining if diameter distributions are reflective of specified structural diversity targets). The addition of structural yield prediction to commonly used density management decision-support aids, as exemplified by this study for SDMDs and by Gove (2004) for traditional stocking guides, enables forest managers to address structural-dependent objectives, while using familiar tools. The structural stocking guide developed by Gove (2004) for eastern white pine (Pinus strobus L.) stands, allows users to recover the underlying diameter frequency distribution at any point in the basal area—density plane thus facilitating silviculture decision-making. Additionally, the diameter distribution information when combined with a diameter–crown area relationship could be used to more precisely define some of the principal relationships within the stocking guide, such as the B-line (Gove, 2004). The structural SDMD presented in this study, enables forest managers to estimate the number of trees in each discrete diameter class at any point during stand development. This information provides the basis for deriving optimal density management regimes in terms of maximizing product value given the intrinsic relationship between diameter and monetary value. Consequently, the structural SDMD and its algorithmic analogue should be of utility as boreal forest management objectives shift from maximizing volumetric yield to optimizing product value. 3.4. Concluding notes The results of this study reconfirms the utility of extending existing stand-level yield models (SDMDs) to stand-level diameter distribution yield models via the parameter prediction approach employing the three-parameter Weibull PDF. This inference has

important practical implications for forest managers within the Canadian Boreal Forest given the large the number of SDMDs already in use and hence potentially expandable. The SDMDMID software provides forest managers with a decision-support tool for density management within black spruce plantations, in terms of deriving optimal density management regimes based on maximizing productivity, product value or optimal site occupancy. Research efforts are continuing in terms of (1) explicitly quantifying the relationship between product value and tree and stand characteristics for black spruce plantations, and (2) assessing the predictive accuracy of the model.

Acknowledgments The authors express their appreciation to Scott Jones, Coordinator, Ontario Terrestrial Assessment Program, Ontario Ministry of Natural Resources for data access, and to the Natural Sciences and Engineering Research Council of Canada for fiscal support (Strategic Project Research Grant 234774).

References ´ lvarez, J.G., Schro¨ der, J., Rodrı´guez, R.S., Ruı´z, A.D., 2002. A Modelling the effect of thinning on the diameter distribution of even-aged Maritime pine stands. For. Ecol. Manage. 165, 57– 65. American National Standards Institute (ANSI), 1978. FORTRAN standard. Publication X3.9, 36 pp. Ando, T., 1962. Growth analysis on the natural stands of Japanese red pine (Pinus densiflora Sieb. et. Zucc.). II. Analysis of stand density and growth (in Japanese; English summary), no. 147. Government of Japan, Bulletin of the Government Forest Experiment Station. Tokyo, Japan, 77 pp. Bailey, R.L., Dell, T.R., 1973. Quantifying diameter distributions with the Weibull function. For. Sci. 19, 97–104. Bailey, R.L., Burgan, T.M., Jokela, E.J., 1989. Fertilized midrotation-aged slash pine plantations-stand structure and yield prediction models. South. J. Appl. For. 13, 76–80. Brooks, J.R., Borders, B.E., Bailey, R.L., 1992. Predicting diameter distributions for site-prepared loblolly and slash pine plantations. South. J. Appl. For. 16, 130–133. Brunsdon, B., 2000. A stand-level perspective of IFM in New Brunswick. In: Bell, F.W., Pitt, D.G., Irvine, M., Parker, W.C., Buse, L.J., Stocker, N., Towill, W.D., Chen, H., Pinto, F., Brown, K., DeYoe, D., McDonough, T., Smith, G., Weber, M.

P.F. Newton et al. / Forest Ecology and Management 209 (2005) 181–192 (Eds.), Intensive Forest Management in Ontario—Summary of a 1999 Science Workshop. Science Development and Transfer Series 003. Government of Ontario, Ministry of Natural Resources, Sault Ste. Marie, Ontario, Canada, p. 11. Burk, T.E., Mack, T.J., 2002. User’s Manual for Resinosa: An Interactive Density Management Diagram for Red Pine in the Lake States. Staff Paper Series Number 158. Department of Forest Resources, College of Natural Resources and Minnesota Agricultural Experiment Station, University of Minnesota, St. Paul, Minnesota, 25 pp. Canadian Council of Forest Ministers (CCFM), 2002. Compendium of Canadian Forestry Statistics [online]. Available at WWWURL: http://nfdp.ccfm.org/frames_news.htm. Clutter, J.L., Bennett, F.A., 1965. Diameter distribution in old-field slash pine plantations. Ga. For. Res. Counc. Rep. 13. Clutter, J.L., Fortson, J.C., Pienaar, L.V., Brister, G.H., Bailey, R.L., 1983. Timber Management: A Quantitative Approach. John Wiley and Sons, New York, USA. Clutter, J.L., Harms, R.W., Brister, G.H., Rheney, J.W., 1984. Stand structure and yields of site-prepared loblolly pine plantations in the lower coastal plain of the Carolinas, Georgia, and north Florida. USDA For. Serv. Gen. Tech. Rep. SE-27. Dell, T.R., Feduccia, D.P., Campbell, T.E., Mann W.F., Jr., Polmer, B.H., 1979. Yields of unthinned slash pine plantations on cutover sites in the West Gulf region. USDA For. Serv. Res. Pap. SO-147. Drew, T.J., Flewelling, J.W., 1977. Some recent Japanese theories of yield-density relationships and their application to Monterey pine plantations. For. Sci. 23, 517–534. Drew, T.J., Flewelling, J.W., 1979. Stand density management: an alternative approach and its application to Douglas-fir plantations. For. Sci. 25, 518–532. Erdle, T., Forest level effects of stand level treatments: using silviculture to control the AAC via the allowable cut effect [online]. In: Newton, P.F. (Ed.) Proceedings of the Expert Workshop on the Impact of Intensive Forest Management on the Allowable Cut. Canadian Ecology Centre, Mattawa, Ontario, November 1–2, 2000. Available at WWW-URL: ftp://ftp.glfc.forestry.ca/pnewton/outgoing/EXPWSF.DOC, pp. 19–30. Feduccia, D.P., Dell, T.R., Mann, W.F., Jr., Campbell, T.E., Polmer, B.H., 1979. Yields of unthinned loblolly pine plantations on cutover sites in the West Gulf region. USDA For. Serv. Res. Pap. SO-148. Gove, J.H., 2004. Structural stocking guides: a new look at an old friend. Can. J. For. Res. 34, 1044–1056. Hagner, M., 2000. Current forest management trends in Scandinavia. In: Bell, F.W., Pitt, D.G., Irvine, M., Parker, W.C., Buse, L.J., Stocker, N., Towill, W.D., Chen, H., Pinto, F., Brown, K., DeYoe, D., McDonough, T., Smith, G., Weber, M. (Eds.), Intensive Forest Management in Ontario—Summary of a 1999 Science Workshop. Science Development and Transfer Series 003. Government of Ontario, Ministry of Natural Resources, Sault Ste. Marie, Ontario, Canada, pp. 9–10. Honer, T.G., Ker, M.F., Alemdag, I.S., 1983. Metric timber tables for the commercial tree species of central and eastern Canada. Information Report M-X-140. Government of Canada, Depart-

191

ment of Agriculture, Canadian Forestry Service, Maritimes Forest Research Centre, Fredericton, New Brunswick. Hyink, D.M., Moser Jr., J.W., 1983. A generalized framework for projecting forest yield and stand structure using diameter distributions. For. Sci. 29, 85–95. Jack, S.B., Long, J.N., 1996. Linkages between silviculture and ecology: an analysis of density management diagrams. For. Ecol. Manage. 86, 205–220. Ker, M.F., Bowling, C., 1991. Polymorphic site index equations for four New Brunswick softwood species. Can. J. For. Res. 21, 728–732. Kilkki, P., Pa¨ ivinen, R., 1986. Weibull function in the estimation of the basal area dbh-distribution. Silva Fenn. 20, 149–156. Kilkki, P., Maltamo, M., Mykkanen, R., Pa¨ ivinen, R., 1989. Use of the Weibull function in estimating the basal area dbh-distribution. Silva Fenn. 23, 311–318. Knoebel, B.R., Burkhart, H.E., Beck, D.E., 1986. A growth and yield model for thinned stands of yellow-poplar. For. Sci. Monogr. 27. Knowe, S.A., 1992. Basal area and diameter distribution models for loblolly pine plantations with hardwood competition in the piedmont and upper coastal plain. South. J. Appl. For. 16, 93–98. Leduc, D.J., Matney, T.G., Belli, K.L., Baldwin, V.C., Jr., 2001. Predicting diameter distributions of longleaf pine plantations: a comparison between artificial neural networks and other accepted methodologies. USDA For. Serv. Res. Pap. SRS-25. Lindsay, S.R., Wood, G.R., Woollons, R.C., 1996. Stand table modeling through the Weibull distribution and usage of skewness information. For. Eco. Manage. 81, 19–23. Little, S.N., 1983. Weibull diameter distributions for mixed stands of western conifers. Can. J. For. Res. 13, 85–88. Liu, C., Zhang, S.Y., Lei, Y., Newton, P.F., Zhang, L., 2004. Evaluation of three methods for predicting diameter distributions of black spruce (Picea mariana) plantations in central Canada. Can. J. For. Res. 34, 2424–2432. Lohrey, R.E., Bailey, R.L., 1976. Yield tables and stand structure for unthinned longleaf pine planatations in Louisiana and Texas. USDA For. Serv. Res. Pap. SO-133. Mabvurira, D., Maltamo, M., Kangas, A., 2002. Predicting and calibrating diameter distributions of Eucalyptus grandis (Hill) Maiden plantations in Zimbabwe. New For. 23, 207–223. Magnussen, S., 1986. Diameter distributions in Picea abies described by the Weibull model. Scand. J. For. Res. 1, 493–502. Maltamo, M., 1997. Comparing basal area diameter distributions estimated by tree species and for the entire growing stock in a mixed stand. Silva Fenn. 31, 53–65. Maltamo, M., Puumalainen, J., Pa¨ ivinen, R., 1995. Comparison of beta and Weibull functions for modeling basal area diameter distribution in stands of Pinus sylvestris and Picea abies. Scand. J. For. Res. 10, 284–295. McTague, T.G., Bailey, R.L., 1987. Compatible basal area and diameter distribution models for thinned loblolly pine plantations in Santa Catarina, Brazil. For. Sci. 33, 43–51. Matney, T.G., Sullivan, A.D., 1982. Compatible stand and stock tables for thinned and unthinned loblolly pine stands. For. Sci. 28, 161–171.

192

P.F. Newton et al. / Forest Ecology and Management 209 (2005) 181–192

Newton, P.F., 1992. Base-age invariant polymorphic site index curves for black spruce and balsam fir within central Newfoundland. North. J. Appl. For. 9, 18–22. Newton, P.F., 1997. Stand density management diagrams: review of their development and utility in stand-level management planning. For. Ecol. Manage. 98, 251–265. Newton, P.F., 2003. Stand density management decision-support program for simulating multiple thinning regimes within black spruce plantations. COMPAG 38, 45–53. Newton, P.F., 2005. Forest production model for upland black spruce stands – optimal site occupancy levels for maximizing net production. Ecol. Mod., in press. Newton, P.F., Weetman, G.F., 1993. Stand density management diagrams and their utility in black spruce management. For. Chron. 69, 421–430. Newton, P.F., Weetman, G.F., 1994. Stand density management diagram for managed black spruce stands. For. Chron. 70, 65–74. Newton, P.F., Lei, Y., Zhang, S.Y., 2004. A parameter recovery model for estimating black spruce diameter distributions within the context of a stand density management diagram. For. Chron. 80, 349–358. Payandeh, B., Wang, Y., 1995. Comparison of the modified Weibull and Richards growth function for developing site index equations. New For. 9, 147–155. Pre´ gent, G., Bertrand, V., Charette, L., 1996. Tables pre´ liminaires de rendement pour les plantations d’Epinette noire au Que´ bec. Me´ moire de recherche forestie`re no. 118. Gouvernement du Que´ bec, Ministe`re des Ressources naturelles, Direction of recherche forestie`re. Rennolls, K., Geary, D.N., Rollinson, T.J.D., 1985. Characterizing diameter distributions by the use of the Weibull distribution. Forestry 58, 57–66. Schreuder, H.T., Swank, W.T., 1974. Coniferous stands characterized with the Weibull distribution. Can. J. For. Res. 4, 518–523. Schreuder, H.T., Hafley, W.L., Bennett, F.A., 1979. Yield prediction for unthinned natural slash pine stands. For. Sci. 25, 25–30. Siipilehto, J., 1999. Improving the accuracy of predicted basal –area diameter distribution in advanced stands by determining stem number. Silva Fenn. 33, 281–301. Smalley, G.W., Bailey, R.L., 1974a. Yield tables and stand structure for loblolly pine plantations in Tennessee, Alabama, and Georgia highlands. USDA For. Serv. Res. Pap. SO-96.

Smalley, G.W., Bailey, R.L., 1974b. Yield tables and stand structure for shortleaf pine plantations in Tennessee, Alabama, and Georgia highlands. USDA For. Serv. Res. Pap. SO-97. Smith, N.J., 1989. A stand-density control diagram for western red cedar, Thuja plicata. For. Ecol. Manage. 27, 235–244. Statistical Analysis System (SAS), 1999. SAS/OR User’s Guide: Mathematical Programming SAS OnlineDoc. SAS Institute Inc., Cary, NC, USA (http://v8doc.sas.com). Sturtevant, B.R., Bissonette, J.A., Long, J.N., 1996. Temporal and spatial dynamics of boreal forest structure in western Newfoundland: silvicultural implications for marten habitat management. For. Ecol. Manage. 87, 13–25. Terry, T., 2000. IFM: a Weyerhaeuser perspective. In: Bell, F.W., Pitt, D.G., Irvine, M., Parker, W.C., Buse, L.J., Stocker, N., Towill, W.D., Chen, H., Pinto, F., Brown, K., DeYoe, D., McDonough, T., Smith, G., Weber, M. (Eds.), Intensive Forest Management in Ontario—Summary of a 1999 Science Workshop. Science Development and Transfer Series 003. Government of Ontario, Ministry of Natural Resources, Sault Ste. Marie, Ontario, Canada, pp. 11–12. Weetman, G.F., 1997. The connection between density management and annual allowable cut. In: Barnsey, C. (Ed.), Proceedings of the Stand Density Management Planning and Implementation Conference, November 6-7, 1997. Edmonton, Alberta, Clear Lake Ltd. Edmonton, Alberta, Canada, pp. 1–3. Weibull, W., 1951. A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297. Zellner, A., 1962. An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias. JASA 57, 348–368. Zhang, S.Y., Chauret, G., 2001. Impact on initial spacing on tree and wood characteristics, product quality and value recovery in black spruce (Picea mariana). Canadian Forest Service Report No. 35/Project No. 1944. Forintek Canada Corporation, SainteFoy, Quebec, 47 pp. Zhang, S.Y., Corneau, Y., Chauret, G., 1998. Impact of precommercial thinning on tree and wood characteristics, product quality, and value in balsam fir. Canadian Forest Service No. 39, Forintek Canada Corp., Sainte-Foy, Quebec, 74 pp. Zhang, S.Y., Chauret, G., Ren, H.Q., Desjardins, R., 2002. Impact of initial spacing on plantation black spruce lumber grade yield, bending properties and MSR yield. Wood Fibre Sci. 34, 460– 475.