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Silva Balcanica, 13(1)/2012

A tentAtive dynAmic site index model for scots pine (Pinus sylvestris) plAntAtions in BulgAriA

Tatiana StankovaForest Research Institute of Bulgarian Academy of Sciences, Bulgaria

Ulises Diéguez-ArandaDepartment of Agroforestry Engineering,

University of Santiago de Compostela, Spain

Abstract

The site index tables for Scots pine (Pinus sylvestris L.) plantations applied in the forest inventory and management in Bulgaria nowadays consist of rela-tive site index classes based on the mean height. They have limited applicability because of their narrow age-height range and poor fit to the actual growth trajec-tories. The main objective of this investigation is to derive a tentative dynamic, base-age invariant site index model for Scots pine plantations in Bulgaria based on the inventory of permanent sample plots, which would allow simple site in-dex estimation and height growth prediction. Five of the most often used growth functions for modeling site index curves (Bertalanffy-Richards, Lundqvist-Korf, Hossfeld, Schumacher and Gompertz) have been tested in all their algebraic difference forms (ADA models), in several published generalized algebraic dif-ference forms (GADA models) and a generalized algebraic difference form of Schumacher’s 2-parameter model was proposed and examined in the present study as well. The candidate regression models, expanded with a first-order autoregressive error structure, were fitted using the base-age invariant dummy variable method. Lundqvist-Korf’s growth function estimated in its anamorphic algebraic difference form was derived to characterize the site quality indices of Scots pine plantations in Bulgaria. The available data from plantation-grown Scots pine in Bulgaria had some limitations and model re-fitting on a data set expanded by additional measurements and stem analysis data is recommended. The study confirmed the reliability of the difference equation approach which can be widely employed for dynamic site index modeling of the forest stands in Bulgaria.

Key words: site quality curves, dynamic site index modeling, difference equation method, base-age invariant techniques, Scots pine plantations

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introduction

The site index models applied in the growth and yield tables in Bulgaria so far (Nedyalkov et al., 1983, Krastanov, Raykov, 2004) are based on a graphic-statistics method introduced (Perthuis de Laillevault 1803 after Pretzsch, 2009), popularized and initially applied (Baur 1876 after Pretzsch, 2009) in western Eu-rope and known as “band method” (Assmann 1970), “strip method” (Pretzsch, 2009) or “sectioning” (Gadow, Hui, 1999). The resultant site index models are of the disjoint polymorphic type (Gadow & Hui 1999) such as the currently imple-mented site quality curves for Scots pine plantations (Krastanov et al., 1983) and Austrian black pine plantations (Tsakov, 1983). An original Bulgarian method was proposed in the 1960-s by Douhovnikov (1965): the intrinsic growth rate indices or the so called “natural indices”. The method derives a coefficient which characterizes the specific growth trend of a tree or a stand based on its growth dynamic in time and is used to classify the stands by their type of growth mode (curve shape) into 1 to 3 main categories, for each of which a set of anamorphic site index curves, around the estimated guiding curve, is developed. The result-ing complex site index models are of the polymorphic non-disjoint curve type (Clutter et al., 1983, Markoff, 2007) and such models have been established for Norway spruce and White fir stands (Douhovnikov, Iliev, 1983), for Scots pine stands (Douhovnikov, Iliev, 1983, Mihov, 1983), coppice beech stands (Tonchev, 2007), and high forest beech stands (Petrin et al., 2010). In spite of the validated high precision and intrinsically dynamic nature of the method, its application in its original form has been abandoned due to difficult practical implementation, requiring preliminary determination of the stand growth mode through indirect indices based on historical records (e.g. dominant stand height at age 10 years, height increment for a particular growth period, age of achieving a particular height) (Mihov, 1986). The method has been gradually simplified to development of anamorphic site index model for one average growth mode, and such approach has been applied to derive site index models for Norway spruce, White fir, Scots pine and Macedonian pine stands (Mihov et al., 2009), oak and beech stands (Mihov et al., 2010).

The most important desirable attributes of the site index equations, which depend on the construction method and the mathematical function, used to de-velop the curves are: polymorphism, sigmoid growth pattern with an inflexion point, horizontal asymptote at old ages or multiple asymptotes (i.e. asymptote varying with site index), logical behavior and base-age invariance (Diéguez-Aran-da et al., 2005a). Among the three main methods described by Clutter et al. (1983) for construction of site index curves, the disadvantages of the guide curve method (producing either anamorphic – in its classical way of application – or poly-morphic site index curves and highly dependent on the representativeness of the data set) and of the parameter prediction method (resulting in site quality model

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dependent on the chosen base age) have imposed the predominant application of the difference equation method (Borders et al. 1984, Parresol, Vissage, 1998, Fontes et al., 2003, Álvarez González et al., 2005). The difference equation ap-proach is flexible for its ability to produce both anamorphic and polymorphic site index curves and the constructed site index models are dynamic, either base-age specific or base-age invariant. It has been developed in two forms: the algebraic difference approach (ADA) proposed by Bailey, Clutter (1974), which allows one of the equation parameters to vary between the sites, and the generalized algebraic difference approach (GADA) originally derived by Cieszewski, Bella (1989) and formally named in this way by Cieszewski, Bailey (2000). The main advantage of the GADA is that one can expand the base equations according to various theories about growth characteristics (e.g., asymptote, growth rate) (Diéguez-Aranda et al., 2005b). For some growth models such as Bertalanffy-Richards and Lundqvist-Korf growth functions GADA is the only way to allow more than one parameter to be site-specific, and let the derivation of more flexible dynamic equations, i.e. of concurrent polymorphism and multiple asymptotes (Diéguez-Aranda et al., 2005b). Both base age specific (BAS) and base age invariant (BAI) methods can be applied to fit equations in algebraic difference form (Cieszewski et al. 2000). The former requires an arbitrary choice of base age prior to fitting the model, and then forcing the model through the chosen height/age point (site index). The BAI meth-ods recognize that each measurement is made with error and, therefore, it seems unreasonable to force the model through any given measurement. BAI methods show methodological superiority, because they allow the curve to be fitted to the observed individual trends in the data and there is no need to make any arbitrary choice regarding measurement intervals (Cieszewski et al., 2000).

Permanent sample plots are being established and re-measured by 10-year intervals by the Forest inventory organizations in Bulgaria since the 1970s to study the forest stand growth and productivity, and to examine different man-agement regimes. To make use of the available repeated-measurements data and considering the potential of the difference equation approach for dynamic site index modeling, the derivation of a tentative dynamic, base-age invariant site index model for Scots pine (Pinus sylvestris L.) plantations in Bulgaria was set as the main objective of the present investigation.

mAteriAls And metHods

data setData set used for development of the site index model consists of 193 age-

height measurements obtained from three sub-sets of permanent sample plots in Scots pine plantations measured 2 to 5 times. The first sub-set includes 21 permanent sample plots, which are being established by the Forest inventory or-ganizations since the 1970s to study the plantation growth and productivity, and

8

for examining of different management regimes. The plots are established fol-lowing the requirement of minimum 200 stems per plot and in 1 to 3 replications of size 400-2000 m2, in 15 to 50-year-old Scots pine plantations, situated in the mountains of Bulgaria at altitudes from 450 to 1600 m a.s.l. The plots have been measured 2 to 4 times every 10 years. The second sub-set is composed of 4 experi-mental plots of 150-300 m2 (87 - 129 trees per plot at the time of plot installation) in Scots pine plantations against erosion (Marinov 1999, 2002, 2006), established at plantation age 14-23 years at altitudes 800-900 m a.s.l. and re-measured 1 to 5 times by 2-3 year periods. The third sub-set includes 7 permanent sample plots, of more than 200 trees each, used for development of growth and productivity tables for Scots pine plantations (Krastanov et al. 1980). The plots are installed in 4 replications, each of 2000 m2 and are established at plantation age 15-45 years and re-measured once by 3-6 year periods. The plots are situated in the moun-tains Vitosha and the Rhodopes at altitudes 800-1550 m a.s.l. Only plots that have not been thinned, or have been exposed to thinnings from below are included in the data set. The plot dominant height was estimated from the experimental mean plot height by the relationship:

Hdom=0.996Hmean+1.455 Stankova et al. (2006).

The descriptive statistics of the dominant heights in the experimental data set are shown in Table 1.

table 1 Descriptive statistics of the dominant heights (m) used for the site index modeling by age classes

Age class (years)

number of measure-

ments

Average minimum maximum standard deviation

10 - 20 22 8.4 6.3 12.2 1.6

20 - 30 31 14.0 9.4 18.3 2.2

30 - 40 60 16.7 12.5 23.4 2.8

40 - 50 44 19.0 15.3 25.9 3.2

50 - 60 19 22.2 16.6 30.6 4.1

60 - 70 13 24.6 20.4 32.5 4.0

70 - 80 4 22.4 21.4 23.4 1.2

growth models and estimation methodNumerous models have been tested in both their integral (H=f(A)) and

difference (H2=f(H1, A1, A2)) form for modeling site quality curves. The most of-ten used growth functions for modeling site index curves (Bertalanffy-Richards, Lundqvist-Korf, Schumacher, Gompertz and Hossfeld) have been considered in

9

the present study. All possible algebraic difference equation forms (ADA models) were examined for each of the functions, several GADA forms derived on the basis of Bertalanffy-Richards, Gompertz and Hossfeld models (Diéguez-Aranda et al., 2005b, Cieszewski et al., 2007) were tested and a generalized algebraic dif-ference equation form of the 2-coefficient Schumacher model was also proposed and examined in the present study. The GADA formulation of Schumacher mod-el considers both coefficients (a and b) as functions of the unknown site specific factor X (Eqs.2 and 3) as follows:

H = aexp(b/A) Eq. (1)a = exp(X) Eq. (2)b = c0 /X Eq. (3)

where A and H denote age (years) and height (m), c0 is a global model pa-rameter and Eq. (1) is the integral equation form of the Schumacher’s model.

After substitution of Eqs. 2 and 3 in Eq. 1 for A=A0 and H=H0, the site specific factor X is derived as:

Eq. (4)

and the generalized algebraic difference equation form of the Schumach-er’s model is:

Eq. (5)

The base-age invariant (BAI) dummy variable approach proposed by Cieszewski et al. (2000) was employed to fit the height growth models in the pres-ent study.

model comparison and base age selectionThe models comparison was preceded by adequacy examination of the bio-

logical properties of the fitted functions. The developed site index models were further compared and the selection of the best site index model was based on several criteria: the adjusted coefficient of determination (R2

adj) and root mean square error (RMSE) of the models, and bias and root mean square error (RMSE) estimated by age classes. The practical implementation of the site index model re-quires choice of base age, which will result in reliable predictions of height at oth-

10

er ages. An objective criterion for such estimation is the relative error coefficient (RE%) proposed by Diéguez-Aranda et al. (2005b), which consists of predicting the heights within the plots through the observed values of all possible base ages and their correspondent heights, both forward and backward, and comparison of the predictions with the observed heights:

Eq. (6)

where Hi, Ĥi and are the observed, predicted and average values of tree height, respectively; n the number of observations; and p the number of param-eters.

results

We need to point out that the available data possess some limitations: rela-tively small number of sample plots, indirect estimation of the dominant stand height, lack of data from stem analysis, which inevitably would impede the fitting of the more complicated, but more flexible and biologically adequate GADA formulations. Some of the tested ADA models and all GADA models but Eq. (5) were discarded in the first steps of the analyses, because of their poor perfor-mance (statistically insignificant global model parameters, lack of model con-vergence, relatively low coefficient of determination) when applied to the data. All model formulations based on the Bertalanffy-Richards model were excluded because they all produced biologically inadequate shape parameter estimates. The models remained for further examination and comparison (Eqs. 7 - 12) are presented in Table 2.

Three anamorphic model formulations, two single asymptote polymorphic model forms and the newly proposed GADA model of concurrent polymorphism and multiple asymptotes were fitted successfully to the data, with coefficients of determination over 0.9 and root mean square errors between 1.23 and 1.41 m (Table 2). Visual examination for errors autocorrelation revealed trends of re-siduals on the lag-residuals from previous observations on the same plot and sug-gested the continuous autoregressive error structure (CAR) modeling to improve the statistical properties of the model. Considering the irregular, unbalanced and longitudinal character of the data in the present study, the error term was ex-panded as exemplified by Diéguez-Aranda et al. (2005b, 2006a):

Eq. (13)

11

tabl

e 2

Sele

cted

mod

els

with

the

ir p

aram

eter

est

imat

es a

nd g

oodn

ess-

of-f

it st

atis

tics

Mod

el f

orm

Mod

el fi

t

Equ

atio

n nu

mbe

r an

d ab

-br

evia

tion*

Inte

gral

for

mD

iffe

renc

e fo

rmP

aram

eter

Est

i-m

ate

Stan

dard

E

rror

Adj

. R2

RM

SE

(m)

12

34

56

7

Eq.

(7)

Sch

u-m

ache

r G

AD

A

c0

-83.

023

4.45

50.

927

1.38

5

r-0

.684

0.25

8

Eq.

(8)

Sch

u-m

ache

r A

DA

_a

b-2

3.70

71.

051

0.93

11.

350

r-0

.686

0.25

3

Eq.

(9)

Lun

dqvi

st-K

orf

AD

A_a

b

6.63

10.

674

0.94

21.

231

c0.

342

0.13

0

r-0

.660

0.25

6

Eq.

(10

)H

ossf

eld

AD

A_a

b

75.0

8313

.679

0.94

21.

230

c1.

012

0.14

2

r-0

.661

0.25

2

12

tabl

e 2

(Con

tinue

d)

Mod

el f

orm

Mod

el fi

t

Equ

atio

n nu

mbe

r an

d ab

-br

evia

tion*

Inte

gral

for

mD

iffe

renc

e fo

rmP

aram

eter

Est

i-m

ate

Stan

dard

E

rror

Adj

. R2

RM

SE

(m)

12

34

56

7

Eq.

(11

)B

ruch

wal

d et

al.

(200

0) A

DA

_a

afte

r H

ossf

eld’

s m

odel

b

41.3

871.

719

0.92

41.

412

r-0

.660

0.29

4

Eq.

(12

)G

ompe

rtz

AD

A_c

a34

.831

1.90

20.

936

1.29

4

b1.

953

0.07

5 A

bbre

viat

ions

: A a

nd H

den

ote

age

(yea

rs)

and

heig

ht (

m),

res

pect

ivel

y, in

the

inte

gral

equ

atio

n fo

rms;

whi

le H

1 is

the

heig

ht a

t ag

e A

1, H

2 is

the

heig

ht a

t ag

e A

2 in

the

diff

eren

ce e

quat

ion

form

s; a

, b

and

c ar

e th

e m

odel

glo

bal p

aram

eter

s, w

hile

r is

the

aut

oreg

ress

ive

para

met

er.

* T

he le

tter

s a,

b a

nd c

in th

e m

odel

abb

revi

atio

n de

note

the

para

met

er o

f the

inte

gral

equ

atio

n fo

rm w

hich

is le

ft fr

ee in

the

resp

ectiv

e di

ffer

ence

eq

uatio

n fo

rm.

13

fig. 1. Root Mean Square Error (RMSE) and Bias for the selected models by age classes

where eij is the jth residual on the ith plot, d=1 for j>1 and d=0 for j=1, r is the first-order autoregressive parameter, tij-tij-1 is the time distance separating the jth from the jth-1 observation within each plot, and εij are independent and identically distributed errors.

The anamorphic ADA form of Hossfeld’s model (Eq. 10), which showed the

14

fig. 2. Graphical examination of the 4 selected models estimated for site index classes 17, 21 and 25 m at base age 50 years

fig. 3. Graphical examination and compari-son of the newly derived and the existing site index models for Scots pine in Bulgaria. a. Relative errors (RE%) in height predictions by age classes for the derived site index mod-el by Lundqvist-Korf; b. Relative site indi-ces by Krastanov et al. (1983) for Scots pine plantations; c. Comparison of the trend of the new model with the trends of the three main growth types for Scots pine stands by Mihov (1983)

comparison to the other three models. The mean biases of all four models were not

15

best statistical properties revealed inadequacy for growth presentation of forest stands similar to this exhibited by the Bertalanffy-Richards model, which estimat-ed early growth peak (to 5 years of age) followed by gradual decrease. The growth model by Gompertz (Eq. 12), on the other hand, overestimated significantly the tree heights at early ages (5-15 years) and predicted unrealistically small growth increment. These three equations were excluded from further comparisons. The remaining four regression models were further compared through estimation of bias and RMSE by age classes (Fig. 1). The anamorphic model by Lundqvist-Korf, along with best statistical properties (Table 2), revealed also smallest values of the RMSE by age classes and its bias was of notably smaller range around the zero in

significantly different from zero and the residual plots exhibited homoscedasti-cally distributed errors (data not shown). The relative errors (RE%) by age classes were computed to determine the most appropriate reference age and the results showed that the age class 40-50 years results in lowest error value of predictions, followed by age class 50-60 years, after excluding the oldest age class because of the small number of observations (Fig. 3a). On the basis of the above finding and examination of the data range at age 50 years, site quality curves for 17, 21 and 25 m site classes at reference age 50 years were estimated and are shown on Fig. 2 together with the permanent plot data for their approximation, and temporary sample plot data (Stankova, Shibuya 2007) for final graphical examination of the four selected site index models. The anamorphic model by Lundqvist-Korf was confirmed as most appropriate for site quality characterization of Scots pine plantations in Bulgaria.

discussion

Assmann (1970) points out that there should be no doubt that the site qual-ity curves have to be constructed as dynamic, considering the probable height development. The derivations of ADA and GADA became milestones in model-ing dynamic site index curves and various formulations of polymorphic, with multiple asymptotes site index curves are being derived recently for modeling site quality of major tree species (Diéguez-Aranda et al. 2005a, 2005b, 2006a, 2006b, Bravo-Oviedo et al., 2007, Cieszewski et al., 2007, Martin Benito et al., 2008, Nord-Larsen et al., 2009).

In spite of the limitations imposed by the available data set, a dynamic, anamorphic site quality model of high goodness-of-fit and precision was derived. Although the polymorphism of the curves is one of the desired attributes of the site index equations (Diéguez-Aranda et al., 2005a), the polymorphic models which resulted in acceptable regression goodness of fit statistics (Bruchwald et al. (2000) ADA_a and Schumacher GADA – Table 2) in this study were inferior to the anamorphic one (Table 2, Fig. 2). The outcome of the comparison can be

16

explained with the pattern of the fit data set, as well as the relative scarcity of data of high site quality plots at early stages and the lack of data for the old growth stages (Table 1). The model still demonstrated its good predictive ability (Figs. 1, 2a, 3a) and is in agreement with the finding by other studies that a model with multiple asymptotes produces better goodness of fit statistics (Elfving, Kiviste 1997, Corral-Rivas et al., 2004, Diéguez Aranda et al., 2006a, 2006b).

Fig. 3c shows a comparison between the growth trend of the Scots pine plantations and the three main growth types derived for natural Scots pine stands in Bulgaria (Mihov, 1983). The figure shows that the plantation growth is similar to the third type of growth mode, particularly in the late growth stages. This find-ing agrees with the general empirical observation for the Scots pine in Bulgaria that the plantation growth pattern is characterized by a fast early growth, rela-tively early culmination and decrease, unlike the slower, but steady stand growth. The derivation of the Lundqvist-Korf’s model to represent the height growth of Scots pine plantations in Bulgaria agrees also with the results by Mihov (1986) for Scots pine stands who concluded the superiority of the same model, which even slightly exceeded the goodness of fit of the model derived by the “natural indices” method.

The reference age preferred in deriving absolute site indices for the forest stands in Bulgaria is 100 years (Mihov et al., 2010), which more or less satisfies the requirement to be close to the stand rotation age (Clutter et al., 1983). The analysis of the variation in the relative error by age classes in this study, however, showed that an early age of 40-50 years is a more appropriate reference value. This result is in agreement with findings by Diéguez-Aranda et al. (2005b, 2006a, b) on the best base age for Pinus radiata and Pinus taeda plantations and for Betula pubescens stands, and agrees with their conclusion that the reference age has to be selected as young as possible, in order to help in earlier decision making of the silvicultural treatments to be applied to the stand.

The site index model is an important part of stand-level growth models (Gadow, Hui, 1999). Even the Stand Density Management Diagrams (SDMD), which are a type of stand-level models that does not include as their integral part such model, require for their practical application the utilization of a site index model at earlier (Newton, Weetman, 1994) or later stage (Sales Luis, Fonseca, 2004, Bario Anta, Álvarez González, 2005, Long, Shaw, 2005). The site index tables for Scots pine plantations, applied together with the whole-stand tables in the forest inventory and management in Bulgaria nowadays (Krastanov et al., 1983), consist of relative site index classes, based on the mean rather than the dominant height, and cover plantation ages up to 50 years. The examination with permanent and temporary sample plots data showed that they are inapplicable to both stand data classification and stand model utilization, because of their lim-ited range and poor fit to the actual growth trajectories (Fig. 3b).

A common weakness in the goodness of fit of all models tested in this

17

study is the early height growth culmination estimated – within the age intervals 5–10–15 years. This result and the height growth tendency to form a set of ana-morphic curves are largely due to the quality of the available data and re-fitting of the site index model on a data set expanded by additional measurements and stem analysis data is recommended. The present study proved the reliability of the difference equation approach and derived a dynamic, base-age invariant site index model with multiple asymptotes for Scots pine plantations in Bulgaria. The model can be applied for site index estimation through the stand dominant height and age, and can be used for data classification in building stand level models and for the stand model practical implementation in planning the management regimes.

Acknowledgements: This article represents part of the research work under the implementation of the Marie Curie Intra-European Fellowship Project within the 7th European Community Frame-work Programme: “Elaboration of advanced-level models for density management of coniferous and broadleaved even-aged natural stands and plantations in Europe” (PIEF-GA-2009-235039), conducted at the University of Santiago de Compostela, Spain.

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e-mail: tatianastankova@yahoo.come-mail: ulises.dieguez@usc.es