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\def\editors	{\href{mailto://c@mcfns.com} {Editor:~Chris~J.~Cieszewski}}
\def\submit 	{Jan.~11,~2009} 	%Submission date can be different than the issue year \issueyear
\def\accept 	{Aug.~12,~2009} 		%The works should be Accepted & Published in the year of the Current_Issue \issueyear
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\def\citename	{Kiviste} 		%"Author" or "FirstAuthor et al."
\def\citeemail	{Andres.Kiviste@emu.ee} 	% Use later: {\href{mailto://\citeemail}{\citename}}
\def\citeetal	{~et~al.} 		% or {} %for a single author; or 
\author{	{\href{mailto://\citeemail}{Andres \citename}}$^1$, 		% Change only the first name of the first author
		{\href{mailto://kylliki.kiviste@emu.ee}{K\"{u}lliki Kiviste}}$^2$ 		% Complete for other
}\affiliation{	\small\it{$^1$Professor, $^2$Lecturer, {\href{http://mi.emu.ee/147346}{IFRE, Estonian University of Life Sciences, Tartu, Estonia. Ph./FAX:(372)731-3156/3156}}}
}

\def\yourtitle	{{Algebraic Difference Equations for Stand Height, Diameter, and Volume 
Depending on Stand Age and Site Factors for Estonian State Forests}} 				%need double {{ for \\ e.g.: {{Title \\ Subtitle}} 

\def\yourkwords	{growth modeling, algebraic difference equation, height, diameter, volume}
\def\yourabstract{
Algebraic difference equations of stand height, diameter, and volume depending on dominant species and site factors have been explored on the basis of Estonian state forest inventory data. Stand variables such as total age, average height, breast height diameter, volume, origin (naturally regenerated or cultivated), forest site type and dominant species from forest inventory database files of Estonian state forests have been used as initial data for this study. A total of 171 data series of height, diameter and volume on age were calculated as averages of data groups by site type, dominant species, origin, and age classes of 5 years. The Cieszewski and Bella (1989) algebraic difference equation has been used for model construction. First, tree parameters of the Hossfeld function were 
estimated for each of the height, diameter and volume series and relationships between the parameters were later studied. In the final model, dominant tree species, thickness of organic layer of soil, stand origin, height, diameter, and volume at given age were used as input variables. The model is included in the Estonian state forest information system and in several software packages for forest inventory data processing. 
}%----------------------------------------------------------------------

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\title{\Large\bf\uppercase\yourtitle} 
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% Continue with the first Section: 

\section{Introduction}
The republic of Estonia lies in Eastern Europe between latitudes 57$^{o}$30' 
and 59$^{o}$49' N and longitudes 21$^{o}$46' and 28$^{o}$13' E. The total 
area of forestland is 2.27 million ha, i.e. 51.9{\%} of the area of Estonia. 
The volume of growing stock on forestland is 451 million cubic meters and is 
showing a trend to increase. Pine stands have the largest area and growing 
stock (710 thousand ha and 151 million m$^{3})$ while birch stands take 
second place (707 thousand ha and 118 million m$^{3})$ and spruce stands 
take third place (404 thousand ha and 87 million m$^{3})$ (Yearbook Forest 
2004, 2005).

Phytogeographically, Estonia belongs to the northern part of the sub-belt of 
the nemoral coniferous or so-called mixed forests of the Northern 
Hemisphere's temperate zone forest belt (Etverk et al 1995). The soils are 
very diverse due to big differences in parent material and in relief, as 
well as in the length of soil genesis and to a lesser extent in climatic 
conditions. 

As a consequence of all this, forests of Estonia vary on a very large scale: 
there are dark boreal spruce forests with treetops at a height of 40 meters; 
heath pine forests, stunted in growth but full of sunshine; unique alvar 
forests growing on a layer of soil that is only a few centimeters thick and 
lies on a stratum of limestone rock; and wet bog forests on peat layers 
several meters thick. 

An ordinated forest typological classification (L\~{o}hmus 1984) has been 
worked out. The set of Estonian forest site types is presented in Table 1. 
According to the dominant tree species there can be either one or several 
forest types in each site type. 

\begin{table*}
\caption{Estonian forest site types by E. L\~{o}hmus (1984) and thickness of 
organic layer of soil (OHOR).}
\begin{center}
\begin{tabular}{|l|p{36pt}|l|l|p{36pt}|l|}
\hline
Code& 
OHOR \par cm& 
Site type& 
Code& 
OHOR \par cm& 
Site type \\
\hline
LL& 
2& 
\textit{Arctostaphylos-alvar}& 
SL& 
1& 
\textit{Hepatica} \\
\hline
KL& 
1& 
\textit{Calamagrostis-alvar}& 
ND& 
1& 
\textit{Aegopodium} \\
\hline
SM& 
4& 
\textit{Cladonia}& 
SJ& 
15& 
\textit{Dryopteris} \\
\hline
KN& 
5& 
\textit{Calluna}& 
AN& 
10& 
\textit{Filipendula} \\
\hline
SN& 
20& 
\textit{Vaccinium uliginosum}& 
TAN& 
15& 
\textit{Carex-Filipendula} \\
\hline
PH& 
4& 
\textit{Rhodococcum}& 
OS& 
20& 
\textit{Equisetum} \\
\hline
JPH& 
4& 
\textit{Oxalis-Rhodococcum}& 
TR& 
20& 
\textit{Carex} \\
\hline
MS& 
10& 
\textit{Myrtillus}& 
RB& 
50& 
\textit{Raised (oligotrophic) bog} \\
\hline
JMS& 
6& 
\textit{Oxalis-Myrtillus}& 
SS& 
50& 
\textit{Transitional (mesotrophic) bog} \\
\hline
KMS& 
13& 
\textit{Polytrichum-Myrtillus}& 
MDS& 
50& 
\textit{Alder-birch (eutrophic-mesotropic) swamp} \\
\hline
KR& 
20& 
\textit{Polytrichum}& 
LD& 
50& 
\textit{Alder (eutrophic) fen} \\
\hline
JK& 
4& 
\textit{Oxalis}& 
KS& 
50& 
\textit{Drained swamp} \\
\hline
\end{tabular}
\label{tab1}
\end{center}
\end{table*}


Until recent times the forest growth and yield tables of Estonia and its 
nearest neighbors have been used in Estonia to offer predictions of forest 
growth. However, Estonian forests are quite variable and the several growth 
tables of differing quality could not describe this variability well enough. 
Thus there has been a growing need for more general forest growth models.

From the aspect of modeling of Estonian forest growth, stand descriptions of 
state forest inventory are the most reliable data available in the state 
forest databases. Traditionally, state forest inventories take place every 
10 years. During the forest inventory, ocular estimates of most stand 
variables (species composition, site type, stand age, height, diameter, 
volume, etc.) are assessed for each sub-compartment. 

The purpose of the present study is to explore a model for prediction of the 
growth of stand height, diameter and volume using the present state of the 
stand and site variables. 

\section{Materials}
As initial data for modeling, stand records of Estonian state forest 
inventory in 1984-1993 were used (Kiviste 1995, Kiviste 1997). Average 
height, mean squared breast height diameter, and volume of 423,919 stands 
were grouped by dominant tree species (Table 2), forest site type (Table 1), 
stand origin (naturally regenerated or cultivated), and stand age (using 
5-year intervals). Data from very young stands (age below 20 years for 
coniferous and hardwood, and 10 years for deciduous forests), from 
over-matured stands and outliers were excluded before the calculation. 

\begin{table*}
\caption{Maximum and minimum ages for stand selection, number of age-series 
and number of stands used for construction of series by dominant species and 
by stand origin (K -- code of origin used in the model).}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Species (Sp)& 
Origin& 
K& 
Min.\ age& 
Max.\ age& 
No.\ series& 
No.\ stands  \\
\hline
Scots pine& 
Naturally regenerated& 
0& 
20& 
120& 
34& 
151710 \\
\hline
Scots pine& 
Planted& 
1& 
20& 
120& 
28& 
54659 \\
\hline
Norway spruce& 
Naturally regenerated& 
0& 
20& 
100& 
25& 
55730 \\
\hline
Norway spruce& 
Planted& 
1& 
20& 
100& 
11& 
21180 \\
\hline
Silver birch and downy birch& 
Naturally regenerated& 
0& 
10& 
70& 
30& 
118080 \\
\hline
Aspen& 
Naturally regenerated& 
0& 
10& 
60& 
9& 
6557 \\
\hline
Common alder& 
Naturally regenerated& 
0& 
10& 
50& 
15& 
5170 \\
\hline
Grey alder& 
Naturally regenerated& 
0& 
10& 
50& 
9& 
9254 \\
\hline
Common oak& 
Naturally regenerated& 
0& 
20& 
120& 
3& 
822 \\
\hline
Common oak& 
Planted& 
1& 
20& 
120& 
3& 
205 \\
\hline
Common ash& 
Naturally regenerated& 
0& 
20& 
100& 
4& 
552 \\
\hline
\end{tabular}
\label{tab2}
\end{center}
\end{table*}


The minimum and maximum age for stand selection, the number of series and 
the number of stands by dominant species and stand origin are presented in 
Table 2. 

As the result of grouping, a total of 171 age-series of height, diameter, 
and volume were obtained. For illustration, empirical height series of pine, 
spruce, birch, and aspen stands are presented in Figures 1-4.

\begin{figure}[htbp]\vspace{1.2in}
\centerline{\includegraphics[width=3.5in,height=2.4in]{fig1.eps}}\vspace{-1.2in}
\caption{Height series of pine stands by forest site type.}
\label{fig1}
\end{figure}

\begin{figure}[htbp]\vspace{1.2in}
\centerline{\includegraphics[width=3.5in,height=2.4in]{fig2.eps}}\vspace{-1.2in}
\caption{Height series of spruce stands by forest site type.}
\label{fig2}
\end{figure}

\begin{figure}[htbp]\vspace{1.2in}
\centerline{\includegraphics[width=3.5in,height=2.4in]{fig3.eps}}\vspace{-1.2in}
\caption{Height series of birch stands by forest site type.}
\label{fig3}
\end{figure}

\begin{figure}[htbp]\vspace{1.2in}
\centerline{\includegraphics[width=3.5in,height=2.4in]{fig4.eps}}\vspace{-1.2in}
\caption{Height series of aspen stands by forest site type.}
\label{fig4}
\end{figure}



For evaluation of the model, data from the network of forest growth 
permanent sample plots in Estonia were used. The network of permanent sample 
plots was established in 1995--2004. By 2005 the first re-measurement data 
had been obtained from 380 sample plots. Of those, 93 sample plots were 
thinned during the period between measurements. The design and method of 
establishing and measuring permanent sample plots is described by A. Kiviste 
and M. Hordo (2002).

%\clearpage 

\section{Methods}
In forest growth modeling, equations for predicting stand variables, for 
example height, are expressed usually in the general form
\begin{equation}
H = f(A, H_{B},)
\label{eq1}\end{equation}
where: H is height of the stand at age A; and
H$_{B}$ is height of the stand at a base age B (site index).

In case we know stand height H$_{1}$ at age A$_{1}$, then for height 
prediction using equation (1), site index H$_{B}$ should be calculated 
first. Site index H$_{B}$ can be obtained by solving the following equation: 
\begin{equation}
H_{1} = f(A_{1}, H_{B}).
\label{eq2}\end{equation}
In most cases for solving equation (2), iteration methods are necessary. To 
reach a solution of necessary precision a few iteration steps are usually 
enough. However, iteration programming is quite a time-consuming and 
complicated task. In certain cases the iteration method may not converge. 
This disadvantage does not occur in the case of algebraic difference 
equations given in the general form
\begin{equation}
H_{2} = g(A_{1}, H_{1}, A_{2}), 
\label{eq3}\end{equation}
where: H$_{2}$ is the predicted stand height at any age A$_{2}$; and
H$_{1}$ is the known stand height at given age A$_{1}$.

The majority of the stand growth algebraic difference equations have been 
derived on the basis of rather simple growth functions (Clutter et al. 1983, 
Rayner 1991, Rennolls 1993), which are not flexible enough for predicting 
the growth of height, diameter, and volume (Kiviste 1988, Kiviste et al 
2002). Derivation of algebraic difference equations from superior growth 
functions is usually algebraically impossible.

A smart and interesting solution is the algebraic difference equation by 
Cieszewski and Bella (1989) for the Hossfeld growth function. Hossfeld's 
growth function is known in the form of
\begin{equation}
\label{eq1}
{H = }\frac{{b}_{0} }{{1+}\frac{{b}_{1}}{{A}^{{b}_{2} }}},
\end{equation}
where: H is stand height at the age A, and
b$_{0}$, b$_{1}$, and b$_{2}$ are the growth function parameters.

Supposing that the growth function passes a point of base age (B , H$_{B})$, 
the function (\ref{eq1}) can be presented as
\begin{equation}
\label{eq2}
{H = }\frac{{H}_{B} \cdot \left( 
{{1+}\frac{{b}_{1} }{{B}^{{b}_{2} }}} 
\right)}{{1+}\frac{{b}_{1} }{{A}^{{b}_{2} 
}}},
\end{equation}
where: H$_{B}$ is stand height at a base age B (site index), and
b$_{1}$ and b$_{2}$ are growth function parameters.

Cieszewski and Bella (1989) learned that parameters H$_{B}$ and b$_{1}$ of 
the growth function (\ref{eq2}) are inversely proportional.
\begin{equation}
\label{eq3}
{b}_{1} =\frac{\beta }{H_B }.
\end{equation}
Replacing the parameter b$_{1}$ in the equation (\ref{eq2}) with the relation (\ref{eq3}) we 
get
\begin{equation}
\label{eq4}
{H = }\frac{{H}_{B} +\frac{\beta }{B^{b_2 }}}{1+\frac{\beta 
\mathord{\left/ {\vphantom {\beta {H_B }}} \right. 
\kern-\nulldelimiterspace} {H_B }}{A^{b_2}}}.
\end{equation}
Substituting the base age B and site index H$_{B}$ in equation (\ref{eq4}) with the 
variables A$_{1}$ and H$_{1}$ the following algebraic difference equation is 
obtained (Cieszewski {\&} Bella 1989).
\begin{equation}
\label{eq5}
H_2 { = }\frac{{H}_{1} 
+{d+r}}{{2+}\frac{{4}\cdot \beta }{\left( {{H}_{1} 
-d+r} \right)\cdot A_2 ^{b_2 }}},
\end{equation}
where: ${d=}\frac{\beta }{{B}^{b_2 }}$, and
\[
{r=}\sqrt {\left( {{H}_{1} -d} \right)^2+\frac{4\cdot \beta 
\cdot H_1 }{A_1 ^{b_2}}}.
\]

The difference equation (\ref{eq5}) proved to be appropriate for modeling of 
dominant height growth of pine forests in Sweden on the basis of permanent 
sample plot data (Elfving {\&} Kiviste 1997), for modeling of dominant 
height growth of birch forests in Sweden on the basis of tree increment core 
data (Eriksson et al 1997) and in other studies (Trincado et al 2003, 
Kasesalu {\&} Kiviste 2001). These successful experiences encouraged us to 
use the same difference equation (\ref{eq5}) for modeling the Estonian height, 
diameter, and volume series.

\textbf{3.1 Estimating the model parameters. }The algebraic difference 
equation (\ref{eq5}) includes three arguments A$_{1}$, H$_{1}$ and A$_{2}$ and three 
parameters B, $\beta $ and b$_{2}$. To estimate the parameters of a 
difference equation, the stand growth data are usually presented as set of 
intervals {\{}(A$_{1}$, H$_{1})$, (A$_{2}$, H$_{2})${\}}. In previous 
studies the parameter B (base age) was fixed by trial and error (Elfving 
{\&} Kiviste 1997, Eriksson et al 1997, Trincado et al 2003). For Estonian 
data we fixed the value 50 years for base age B. Parameters $\beta $ and 
b$_{2}$ were estimated using the procedure of non--linear regression 
analysis on the interval data. 

In site index models (Cieszewski {\&} Bella 1989, Elfving {\&} Kiviste 1997, 
Eriksson et al 1997) parameters $\beta $ and b$_{2}$ were considered as 
constants for each tree species for a certain geographical region. In that 
case model (\ref{eq5}) presents a one-parameter set of growth curves upon 
the age/height plane. According to our previous studies (Kiviste, 
1995) the growth curves depend on site index and on the site properties 
(thickness of organic layer). Thus the height, diameter and volume should be 
modelled as a two-parameter set of curves. 

In this study we used a combined method. In the first modeling step of, a 
total of 171 age-series of height, diameter, and volume were approximated 
using the three-parameter Hossfeld function (\ref{eq1}). Using the non-linear 
regression procedure NLIN of SAS software (SAS Institute Inc. 1989) a set of 
parameters b$_{0}$, b$_{1}$, and b$_{2 }$were estimated for each height (H), 
diameter (D), and volume (M) series. To distinguish the set of parameters we 
added a letter respectively, for example bH$_{2}$ is the parameter b$_{2}$ 
for mean height. For this model residual standard errors of 0.48 m, 0.66 cm, 
and 12.4 m$^{3}$ha$^{-1}$ were estimated in relation to the height, diameter 
and volume series, respectively. 

The Analysis of Variance proved significant difference of the parameter 
bH$_{2}$ by tree species. The variability of the parameter bH$_{2}$ by tree 
species is represented in the Figure 5. The different values of parameter 
b$_{2}$ by tree species were observed on the box-plots of diameter and 
volume as well. Medians of parameter b$_{2}$ for height, diameter and volume 
were estimated for each species (Table 3). 

\begin{figure*}[htbp]
\centerline{\includegraphics[width=5in,height=3.5in]{Kiviste065.eps}}
\caption{Box-plots of parameter bH$_{2}$ by dominant species. bH$_{2}$ = 
parameter b$_{2}$ for mean height.}
\label{fig5}
\end{figure*}


In the second modeling step, the inversely proportional relation of 
parameters bH$_{1}$, bD$_{1}$ and bM$_{1}$ and variables H$_{50}$, D$_{50}$ 
and M$_{50}$ were studied. The scatter-plot of the parameter bH$_{1}$ 
against the site index H$_{50}$ (Figure 6) shows large variation of the 
parameter bH$_{1}$. Nevertheless, the inversely proportional relationship by 
species could be observed in the scatter-plots for height (Figure 6), 
diameter, and volume.

\begin{figure*}[htbp]
\centerline{\includegraphics[width=5in,height=3.5in]{Kiviste066.eps}}
\caption{Scatter-plot of parameters bH$_{1}$ and site index H$_{50}$ by 
species. bH$_{1}$ = parameter b$_{1}$ for mean height.}
\label{fig6}
\end{figure*}


Next, coefficients $\beta $H, $\beta $D and $\beta $M for each age-series 
were calculated using equation (\ref{eq3}). We studied the relationship of the 
coefficients $\beta $H, $\beta $D and $\beta $M on forest type variables: 
dominant tree species, thickness of organic layer of the soil, drainage and 
the origin of the stand. Among these variables the thickness of organic 
layer is continuous, the other are discrete variables. For the covariance 
analysis, the procedure of general linear methods (GLM) of the SAS software 
(SAS Institute Inc. 1989) was used. In the analysis, every series was 
weighted proportionally with the number of stands and inversely proportional 
with the residual variance calculated at the first step of modeling. Only 
the significant variables (tree species, thickness of organic layer and 
origin of the stand, significance level $\alpha $ = 0.05) were included in 
the model. The origin of the stand was set to 0 when the stand was 
cultivated (seeded or planted) and 1 when the stand was naturally 
regenerated. The following linear model was obtained: 
\begin{equation}
\beta = C_{0} + C_{1}\cdot ln(OHOR + 1) + C_{2}\cdot K,
\label{eq9} \end{equation}
where: OHOR is the thickness of organic layer of soil cm; 
K is the dummy variable (Table 2);
C$_{0}$ is a constant depending on tree species; and
C$_{1}$, C$_{2}$ are other model constants.

\section{Results}
An algebraic difference equation was explored for predicting stand height 
(H$_{2})$, breast height diameter (D$_{2})$ and volume (M$_{2})$ at any age 
(A$_{2})$ on the basis of the present state of stand description data 
(A$_{1}$, H$_{1}$, D$_{1}$, M$_{1})$. The algorithm is the following.

\textbf{1.} Determine the input values of the model:
\begin{itemize}
\item[-]dominant tree species (pine, birch, spruce, aspen, grey alder, common 
alder, oak or ash);

\item[-] thickness of the organic layer of soil OHOR cm (Table 1);

\item[-] origin of the stand K (Table 2);

\item[-] stand age at a given moment A$_{1}$ years;

\item[-] stand height (H$_{1})$ m, diameter (D$_{1})$ cm or volume (M$_{1})$ 
m$^{3}$ha$^{-1}$ at a given moment;
\end{itemize}

\textbf{2. }Find the constants bH$_{2}$, bD$_{2}$, bM$_{2}$, CH$_{0}$, 
CD$_{0}$, and CM$_{0}$ according to the dominant tree species (Table 3).


\begin{table*}[htbp]
\caption{Parameter estimates for height, diameter and volume algebraic 
difference equations.}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Species& 
bH$_{2}$& 
bD$_{2}$& 
bM$_{2}$& 
CH$_{0}$& 
CD$_{0}$& 
CM$_{0}$ \\
\hline
Pine& 
1.58& 
1.33& 
1.93& 
8319& 
6051& 
380544 \\
\hline
Spruce& 
1.71& 
1.54& 
2.20& 
12867& 
9805& 
875924 \\
\hline
Birch& 
1.48& 
1.37& 
2.05& 
4990& 
5034& 
446641 \\
\hline
Aspen& 
1.30& 
1.15& 
1.77& 
3882& 
7092& 
310877 \\
\hline
Common alder& 
1.41& 
1.41& 
1.93& 
4228& 
4438& 
378317 \\
\hline
Grey alder& 
1.38& 
1.35& 
1.78& 
2749& 
2864& 
205882 \\
\hline
Oak& 
1.61& 
1.45& 
2.02& 
6742& 
10509& 
277948 \\
\hline
Ash& 
1.35& 
1.03& 
2.12& 
3732& 
5405& 
345440 \\
\hline
\end{tabular}
\label{tab3}
\end{center}
\end{table*}

\textbf{3}. Calculate the coefficients {\ss}H, {\ss}D, and {\ss}M of the 
equation (9) (LN is a function of the normal logarithm).

{\ss}H = CH$_{0}$ - 493$\cdot $LN(OHOR + 1) + 1355$\cdot $K; (10)

{\ss}D = CD$_{0}$ - 306$\cdot $LN(OHOR + 1); (11)

{\ss}M = CM$_{0}$ - 54348$\cdot $LN(OHOR + 1) + 56290$\cdot $K. (12)

\textbf{4. }Calculate the variables dH, rH, dD, rD, dM and rM (SQRT is a 
function of the square root): 

dH = {\ss}H/50$^{bH2}$ (13)

rH = SQRT((H$_{1}$ - dH)$^{2}$ + 4$\cdot ${\ss}H$\cdot 
$H$_{1}$/A$_{1}^{bH2})$; (14)

dD = {\ss}D/50$^{bD2}$ (15)

rD = SQRT((D$_{1}$ - dD)$^{2}$ + 4$\cdot ${\ss}D$\cdot 
$D$_{1}$/A$_{1}^{bD2})$; (16)

dM = {\ss}M/50$^{bM2}$ (17)

rM = SQRT((M$_{1}$ - dM)$^{2}$ + 4$\cdot ${\ss}M$\cdot 
$M$_{1}$/A$_{1}^{bM2})$. (18)

\textbf{5. }Calculate the predicted height (H$_{2}$, m), diameter 
(D$_{2}$, cm), and volume (M$_{2}$, m$^{3}$ha$^{-1})$ at desired age 
A$_{2}$.

H$_{2}$ = (H$_{1}$ + dH+ rH)/(2 + 4$\cdot ${\ss}H$\cdot 
$A$_{2}^{-bH2}$/(H$_{1}$ - dH + rH)), (19)

D$_{2}$ = (D$_{1}$ + dD + rD)/(2 + 4$\cdot ${\ss}D$\cdot 
$A$_{2}^{-bD2}$/(D$_{1}$ - dD + rD)), (20)

M$_{2}$ = (M$_{1}$ + dM + rM)/(2 + 4$\cdot ${\ss}M$\cdot 
$A$_{2}^{-bM2}$/(M$_{1}$ - dM + rM)). (21)

\textbf{6.} The algebraic difference model (10)--(21) can also be used for 
site index calculation. In this case the base age of site index (for example 
100 years) should be assigned to argument A$_{2}$.

7. If we know the values of parameters H$_{50}$, D$_{50}$, and M$_{50}$ for 
a certain site type then stand height, diameter, and volume can be predicted 
using the following equations. Average values of parameters H$_{50}$, 
D$_{50}$, and M$_{50}$ for most Estonian forest types are presented in the 
worksheet "Andmed" of MS Excel file (http://www.eau.ee/$\sim 
$akiviste/kktab2.xls).

H = (H$_{50}+\beta $H/50$^{bH2})$/(1+($\beta $H/H$_{50})\cdot 
$A$^{-bH2})$, (22)

D = (D$_{50}+\beta $D/50$^{bD2})$/(1+($\beta $D/D$_{50})\cdot 
$A$^{-bD2})$, (23)

M = (M$_{50}+\beta $M/50$^{bM2})$/(1+($\beta $M/M$_{50})\cdot 
$A$^{-bM2})$. (24)

The difference model (10)--(21) was fitted to the height, diameter, and 
volume series by finding the most suitable values of H$_{50}$, D$_{50}$ and 
M$_{50}$ for each series. Predictions for series were calculated from the 
state A$_{1}$ = 50, H$_{1}$ = H$_{50}$, D$_{1}$ = D$_{50}$ and M$_{1}$= 
M$_{50}$. No significant bias between predictions and height, diameter, and 
volume series were found. The residual standard errors 0.57 m, 0.83 cm, and 
17.0 m$^{3}$ha$^{-1}$ of the model were calculated in relation to the 
height, diameter and volume series. These residual standard errors were 
slightly higher than those in the case of the Hossfeld function (\ref{eq1}). 

\section{Model evaluation}
The algebraic difference model (10)--(21) was evaluated on 287 
permanent-sample plot data measured twice with an interval of 5 years in 
1995--2004. The plots were located randomly in different parts of Estonia 
and the stands were not thinned between the two measurements. Also, plots 
with great mortality caused by natural disturbances were excluded from the 
analysis.

Data from the first measurement of plots (stand age, height, diameter, 
volume, thickness of organic layer of soil, stand origin, dominant species) 
were assigned to input variables of the model. Using the difference model, 
stand height, diameter, and volume were predicted five years forward and 
compared with the plot re-measurement data.

In Figures 7, 8, and 9 the differences between the results of the second 
measurements and predicted values for stand height (EH), diameter (ED), and 
volume (EM) are presented. 


\begin{figure*}[htbp]
\centerline{\includegraphics[width=5in,height=3.5in]{Fig7.eps}}
\caption{Errors in meters of the difference model for predicting five-year 
stand height growth, depending on stand age at first measurement. The trend 
curves show model bias and its 95{\%} confidence limits. EH = H$_{2}$actual 
-- H$_{2}$model}
\label{fig7}
\end{figure*}


\begin{figure*}[htbp]
\centerline{\includegraphics[width=5in,height=3.5in]{Fig8.eps}}
\caption{Errors in centimeters of the difference model for predicting 
five-year diameter growth, depending on stand age at first measurement. The 
trend curves show model bias and its 95{\%} confidence limits. ED = 
D$_{2}$actual -- D$_{2}$model.}
\label{fig8}
\end{figure*}


\begin{figure*}[htbp]
\centerline{\includegraphics[width=5in,height=3.5in]{Fig9.eps}}
\caption{Errors in m$^{3}$ha$^{-1}$ of the difference model for predicting 
five-year volume growth, depending on stand age at first measurement. The 
trend curves show model bias and its 95{\%} confidence limits. EM = 
M$_{2}$actual -- M$_{2}$model.}
\label{fig9}
\end{figure*}

No overall bias of height and diameter growth predictions can be observed in 
Figures 7 and 8. The residual standard errors of five-year height and 
diameter growth predictions were 0.86 m and 0.58 cm respectively. However, 
at young ages the model slightly overestimates and at mature ages 
underestimates the actual growth of height and diameter. 

Such trends were not revealed when difference model predictions were 
compared with initial data (height, diameter, and volume series compiled 
from forest inventory data). However, a similar effect became evident when 
site indices of forest inventories in the 1950s and 1990s were compared 
(Kiviste, 1999). This trend could be explained by the hypothesis that forest 
growth conditions were improved and the stand growth was accelerating in 
Estonia during the last decades (Nilson {\&} Kiviste 1986).

Figure 9 shows that volume growth predictions are on an average 20 
m$^{3}$ha$^{-1}$ lower than their actual values by the permanent plot data. 
Apparently, this could be caused by the fact that thinned and seriously 
damaged stands were excluded from the comparison while most Estonian forest 
stand data (including thinned and damaged stands) was used for model 
building. The residual standard error of five-year volume growth predictions 
was 15 m$^{3}$ha$^{-1}$.


\section{Discussion}
In this study, a system of algebraic difference equations for prediction of 
stand height, diameter, and volume of Estonian forests have been explored. 
The model summarizes large amounts of forest inventory data which is its 
major advantage in comparison with previous models and growth and yield 
tables used in Estonia. The model parameters cover a huge variety of forest 
site properties, which enables us to generalize forest growth for different 
forest site types using a smart system of equations. 

The structure of the model expressed in the form of algebraic difference 
equations is a convenient way of using it and enables its easy employment in 
applications. The algebraic difference model (10)--(21) proved to be 
reliable and trouble-free and that is one reason why it is included into the 
Estonian state forest information system and into several software packages 
for forest inventory data processing.

The model (10)--(21) describes most reliably the growth of dominant forest 
types from the age of pole forests up to the age of matured forests. As a 
rule, model extrapolation beyond the range of initial data is not 
recommended. However, the basic function of the model is a classical 
Hossfeld growth function, which is one of the most suitable functions for 
forest growth modeling (Kiviste 1988, Kiviste et al 2002); thus we should 
obtain reasonable extrapolations even for young stands and over-matured 
stands. 

Upon approximation of the height, diameter and volume series, the number of 
stands was used as the weight of each observation. Therefore, the 
regularities of the most widely spread species like pine, spruce, and birch, 
and dominating site types like \textit{Myrtillus, Rhodococcum}, and \textit{Aegopodium} have been ``imposed'' on relatively 
uncommon species and site types. Using the weight function for modeling 
relationships between parameters was necessary because series of rare forest 
types having a relatively low number of observations appeared too 
``erratic'' to detect any regularities.

The set of input parameters of difference model (10)--(21) does not contain 
several influential variables like stand density, stand composition, forest 
management schedules, etc. That is why volume growth in unthinned permanent 
plots was on an average higher than predicted. However, adding new variables 
into the model seems to be quite sophisticated while forest inventory data 
only are used for model building. For building a more detailed stand growth 
model a huge amount of data from a well-designed set of permanent plots 
should be used.

The model is based on Estonian state forest inventory data collected in 
1984--1993, grouped by forest type and age class. Those series express the 
relationship of how conditional average values of height, diameter, and 
volume are depending on stand age. These series can coincide with the real 
growth of the stands (assuming that the ocular estimates of forest surveyors 
are free of systematic errors) only when the growth conditions of the stands 
have been stable in time. Several studies, however, point at changing growth 
conditions, demonstrating a considerable increase in forest growth during 
recent decades (Nilson {\&} Kiviste 1984, Eriksson {\&} Johansson 1993, 
Elfving {\&} Tegnhammar 1996). In that case the model offered in this paper 
will actually give a bit smaller predict than realistic prognoses. 

\section{Conclusion}
A method for construction of an algebraic difference model from forest 
inventory stand description data has been presented in this paper. Using 
this method, system of algebraic difference equations (10)--(21) have been 
explored for predicting stand height, diameter, and volume growth on the 
basis of the present state of stand description data. 

As initial data for this study, stand variables like total age, average 
height, diameter, volume, stand origin, site type and stand composition by 
species from database files of all Estonian state forest districts have been 
used. Height, diameter and volume series on age were calculated as averages 
of data groups by site type, by dominant species, by origin and by age 
classes of 5 years. A total of 171 data series has been created from 423,919 
stand descriptions.

The Cieszewski, Bella (1989) algebraic difference equation (\ref{eq5}) has been used 
for model construction. First, tree parameters of Hossfeld function (\ref{eq1}) were 
estimated for each of the height, diameter and volume series, and later 
relationships between the parameters were studied. 

Finally, an algebraic difference equation model (10)--(21) has been 
developed. Dominant tree species (Table 2), thickness of organic layer of 
soil (Table 1), stand origin (Table 2), height, diameter, and volume at 
given age were used as input variables of the model. Parameter estimates of 
the model are presented in Table 3 and in equations (10)--(12).

No significant bias between model predictions and initial data (height, 
diameter, and volume series) were found. The residual standard errors 0.57 
m, 0.83 cm, and 17.0 m$^{3}$ha$^{-1}$ of the model were calculated in 
relation to the height, diameter and volume series.

The model (10)--(21) was evaluated on data from 287 permanent sample plots 
measured twice with an interval of five years in 1995--2004. The residual 
standard errors of five-year height and diameter increment predictions were 
0.86 m and 0.58 cm respectively. However, at young ages the model slightly 
overestimates and at mature ages underestimates the actual growth of height 
and diameter. Volume growth predictions were on an average 20 
m$^{3}$ha$^{-1}$ lower than their actual values on the basis of the 
permanent plot data. This could be caused by the fact that thinned and 
seriously damaged stands were excluded from the comparison while most 
Estonian forest stand data (including thinned and damaged stands) was used 
for model building. The residual standard error of five-year volume 
increment predictions was 15 m$^{3}$ha$^{-1}$.

The structure of the model expressed in the form of algebraic difference 
equations is a convenient way of using it and enables its easy employment in 
applications. The model (10)--(21) proved to be reliable and trouble-free, 
which is one reason why it is included in the Estonian state forest 
information system and in several software packages for forest inventory 
data processing.

\section*{Acknowledgements}
The study was supported by the Estonian \marginpar{\hspace{-.4in}\tiny{Erratum:}}\marginpar{\hspace{-.4in}\tiny{9/6/09}}Ministry of Education and Research (project SF0170014s08). Thanks are due to the two anonymous reviewers for their help in improving the manuscript.

\section*{References}
\begin{description}

\item Cieszewski, C. J. and I. E. Bella. 1989. Polymorphic height and site index curves for lodgepole pine in Alberta. Can.~J. For.~Res. 19: 1151-1160.

\item Clutter, J. L., J. C. Fortson, L. V. Pienaar, G. H. Brister and R. L. 
Bailey. 1983. Timber management: a quantitative approach. John Wiley {\&} 
sons, New York, 333 p.

\item Elfving, B. and A. Kiviste. 1997. Construction of site index equations for Pinus sylvestris L. using permanent research plot data in Sweden. For.~
Eco.~Man., 98(2): 125-134.

\item Elfving, B. and L. Tegnhammar. 1996. Trends of tree growth in Swedish 
forests 1953-1992. An analysis based on sample trees for the National Forest 
Inventory. Scan.~J. For.~Res., 11: 26-37.

\item Eriksson, H., U. Johansson and A. Kiviste. 1997. A site-index model for pure and mixed stands of Betula pendula and Betula pubescens in Sweden. Scan.~J. For.~Res., 12(2): 149-154.

\item Eriksson, H. and U. Johansson. 1993. Yields of Norway spruce (Picea abies (L.) Karst.) in two consecutive rotations in southwestern Sweden. Plant and Soil 154: 239-247.

\item Etverk, I., K. Karoles, E. L\~{o}hmus, T. Meikar, R. M\"{a}nni, T. Nurk, J. Pikk, T. Randveer, \"{U}. Tamm, U. Veibri and A. \"{O}rd. 1995. Estonian forests and forestry. Tallinn. 128 p.

\item Kasesalu H. and A. Kiviste. 2001. The Kuril larch (Larix Gmelinii 
var. Japonica (Regel) Pilger) at J\"{a}rvselja. Baltic Forestry 7(1): 59-66.

\item Kiviste, A. 1988. Forest growth functions. Tartu, 108 + 171 p. (in Russian)

\item Kiviste, A. 1995. Eesti riigimetsa puistute k\~{o}rguse, diameetri ja 
tagavara s\~{o}ltuvus puistu vanusest ja kasvukohatingimustest 1984.-1993.a. 
metsakorralduse takseerkirjelduste andmeil. Eesti 
P\~{o}llumajandus\"{u}likooli teadust\"{o}\"{o}de kogumik. 181. Tartu, 
132-148.

\item Kiviste, A. 1997. Eesti riigimetsa puistute k\~{o}rguse, diameetri ja 
tagavara vanuseridade diferentsmudel 1984.-1993.a. metsakorralduse 
takseerkirjelduste andmeil = Difference equations of stand height, diameter 
and volume depending on stand age and site factors for Estonian state 
forests. Eesti P\~{o}llumajandus\"{u}likooli teadust\"{o}\"{o}de kogumik. 
189. Tartu, 63-75.

\item Kiviste, A. 1999. Site index change in the 1950s--1990s according to 
Estonian forest inventory data. Causes and consequences of accelerating 
tree growth in Europe. EFI Proceedings 27: 87--100.

\item Kiviste, A., J. G. Alvarez Gonzalez, A. Rojo Alboreca and A. D. Ruiz 
Gonzalez. 2002. Funciones de crecimiento de aplicacion en el ambito 
forestal. Madrid, 190 p. 

\item Kiviste, A. and M. Hordo. 2002. Eesti metsa kasvuk\"{a}igu 
p\"{u}siproovit\"{u}kkide v\~{o}rgustik = Network of permanent forest growth 
plots in Estonia. Metsanduslikud uurimused 37. Tartu, 43--58.

\item L\~{o}hmus, E. 1984. Eesti metsakasvukohat\"{u}\"{u}bid = Estonian forest site types. Tallinn, 88 p.

\addtolength{\textheight}{-6.7truein}

\item Nilson, A. and A. Kiviste. 1984. M\"{a}nnikute ``kasvuk\"{a}igu'' mudel 
t\"{u}piseerimata kasvukohatingimuste j\"{a}rgi = Pine stand growth model in 
continuous site types. Eesti P\~{o}llumajanduse Akadeemia teaduslike 
t\"{o}\"{o}de kogumik 151. Tartu, 50-59.

\item Nilson, A. and A. Kiviste. 1986. Reflection of environmental changes in 
models of forest growth composed using different methods. Monitoring of 
forest ecosystems. Abstracts of scientific conference. Kaunas, 
05-06.06.1986. Kaunas-Academy, 336-337. (In Russian).

\item Rayner, M. E. 1991. Site index and dominant height growth curves for 
regrowth karry (Eucalyptus diversicolor F. Muell.) in south-western 
Australia. For.~Eco.~Man., 44: 261-283.

\item Rennolls, K. 1993. Forest height growth modelling. Growth and yield 
estimation from successive forest inventories. Proceedings from the IUFRO 
conference, held in Copenhagen, 14-17 June. 231-238.

\item SAS Institute Inc. 1989. SAS/STAT{\textregistered} User's Quide, Version 6, Fourth Edition, Vol.~2, Cary, NC: SAS Institute Inc., 846 pp.

\item Trincado, G., A. Kiviste and K. von Gadow. 2003. Preliminary site index 
models for native Roble (Nothofagus obliqua) and Rauli (N. alpina) in Chile. 
New Zealand Journal of Forest Science 32(3): 322-333.

\item Yearbook Forest 2004. 2005. Compiled and edited by Metsakaitse-ja 
Metsauuenduskeskus. Tartu, 183 p.

\label{docend}
\end{description}

\end{document}
