2. REVIEW OF LITERATURE Since this thesis is concerned with optimising stand density in teak plantations using growth models, developments associated with important concepts and techniques involved in the subject are reviewed here so as to facilitate later discussion without getting involved into details of the methodology. Other than indicating the current status of the subject field, the review also justifies the selection of the methods from a broad set of alternatives possible. The concepts that are reviewed in this chapter include stand density and its optimisation, growth models, intrinsic biological units for measuring growth, allometric relations, fractal geometry and its application to tree growth. This is followed by a review of past works reported on thinning and rotation age and also environmental effects of growing teak plantations. 2.1. Stand density One of the main goals of plantation forestry is to maximize volume growth, improve wood quality, and thus, increase the returns. The easiest and often profitable way to achieve this goal is to control stand density by initial spacing and subsequently by thinning. Most often foresters measure density by number of trees per unit area. Harper (1977) reported that tree size is an indispensable component of stand density. Although foresters had been more imaginative than other ecologists and had developed a rich variety of density measures (West, 1982) and its kinds (absolute density, relative stand density, stocking and stockability), only one was being commonly used: the basal area. Thus a popular density guide (Gingrich, 1967), used as a Forest Service standard for stocking guide, was based on basal area. Several other variables were also proposed as density measures. They include average distance between trees expressed in proportion to average height or diameter, stand volume, crown closure, and leaf area. Each of these has some advantages and disadvantages. Yet these variables are not satisfactory for the same reason as basal area-stocking (that is, degree of crowding) changes with tree size (Assmann, 1970; West, 1982; Vose and Allen, 1988). 5
Transcript
2. REVIEW OF LITERATURE
Since this thesis is concerned with optimising stand density in teak
plantations using growth models, developments associated with important
concepts and techniques involved in the subject are reviewed here so as to
facilitate later discussion without getting involved into details of the
methodology. Other than indicating the current status of the subject field, the
review also justifies the selection of the methods from a broad set of
alternatives possible. The concepts that are reviewed in this chapter include
stand density and its optimisation, growth models, intrinsic biological units for
measuring growth, allometric relations, fractal geometry and its application to
tree growth. This is followed by a review of past works reported on thinning
and rotation age and also environmental effects of growing teak plantations.
2.1. Stand density
One of the main goals of plantation forestry is to maximize volume growth,
improve wood quality, and thus, increase the returns. The easiest and often
profitable way to achieve this goal is to control stand density by initial spacing
and subsequently by thinning. Most often foresters measure density by
number of trees per unit area. Harper (1977) reported that tree size is an
indispensable component of stand density. Although foresters had been more
imaginative than other ecologists and had developed a rich variety of density
measures (West, 1982) and its kinds (absolute density, relative stand density,
stocking and stockability), only one was being commonly used: the basal area.
Thus a popular density guide (Gingrich, 1967), used as a Forest Service
standard for stocking guide, was based on basal area.
Several other variables were also proposed as density measures. They include
average distance between trees expressed in proportion to average height or
diameter, stand volume, crown closure, and leaf area. Each of these has some
advantages and disadvantages. Yet these variables are not satisfactory for the
same reason as basal area-stocking (that is, degree of crowding) changes with
tree size (Assmann, 1970; West, 1982; Vose and Allen, 1988).
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Despite numerous attempts, finding an optimal density, which maximizes
volume growth of forest stands, has been an elusive task primarily because a
clear definition of density was lacking. Number of trees or basal area does not
make up a valid measure of stand density, as both are incomplete by
themselves to describe the extent of crowding in a stand. For a fixed number
of trees, when the diameter increases, even understocked stands become
overstocked. For constant basal area, the trend of stocking is opposite. Both
basal area and number of trees can be presented as a product of the number
of trees per unit area, N, and mean diameter, D, of the stand with appropriate
choice for the power of D as 2 or 0. The opposite trends in stocking suggests
that there is some power r, confined between 0 and 2, which produces a
proper measure of density, i.e., a measure which remains constant in equally
dense stands, regardless of their diameter. Reineke (1933) identified this
power r and offered an index based on the number of trees and diameter. This
index is a more reasonable measure of density than leaf area because it is a
stable variable that integrates both aboveground and belowground competition
as well as environmental conditions. Reineke’s index (S) is given by
r
DNS
=
4.25 (2.1)
where, N = Number of trees per ha
D = Quadratic mean diameter of trees in cm
r is a parameter which is taken as 1.6 for all practical purposes.
Reineke’s index lumps together two processes of self-thinning viz., increasing
average crown size expressed by diameter and diminishing crown closure.
Zeide (2002) further modified this index to describe both processes explicitly.
The modified index was
)4.25(
4.25
−
= Dc
b
eDNS (2.2)
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where, b = Rate of tree mortality caused by the increase in crown size
c = Mortality rate due to diminishing canopy closure
Equation (2.2) defines stand density as the number of trees per unit area with
D = 25.4 cm. Equation (2.2) applies to any even-aged stand. The definition
(Equation (2.2)) does not impose canopy gaps. If canopy closure is complete, c
would be 0 and Equation (2.2) would be reduced to Equation (2.1).
However, for use in growth equations, the exponential part of the index was
found redundant as this aspect could be taken care of by other growth
parameters, thus reducing the modified index to a form similar to that of
Reineke’s index but with a smaller b which can be interpreted as a measure of
self- tolerance of a species, i.e., the ability of trees to compete with or tolerate
conspecifics (Zeide, 1985). This ability is measured by the proportion of trees
eliminated during the period dt by a certain increase in average diameter, dD/Ddt.
Reduced form of the modified Reineke’s index is being used in this thesis.
2.2. Optimal stand density
Relating growth and density is a principal problem in forestry. Particularly
important is the relationship at which the output of desired products reaches
the maximum. Density at this point is called the optimal density. Search for
this optimum has been conducted since the beginning of forestry without
producing a convincing answer. Even the existence of optimal density was
questioned in the past by many foresters. Curtis et al. (1997) denied the
existence of such a density. On the other hand, Moller (1954) asserted that
frequent thinning of moderate intensity produces the density that maximizes
growth. At present, many foresters believe that volume growth does not
change much over a wide range of stand densities, so that thinning can
redistribute growth from smaller to larger stems but not increase its amount
(Zeide, 2001). One of the reasons is that the effect of density on growth is not
always separated from those of tree size and age. Such a separation is not
easy when the relationship between density and growth is expressed as a
graph (Langsaeter’s curve).
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Zeide (2004) as part of a long derivation proposed the following simple model
that expresses volume growth of an average tree as a function of three
predictors viz., tree size, age and density.
cSqtp eeaDdtdv /' −−= (2.3)
where, dv/dt = Current volume growth of a tree
D = Diameter at breast-height of the tree
t = Age (year)
S = Stand density index (Reineke, 1933)
a, p’, q and c are parameters
It allows one to calculate the density that maximizes volume growth at any
given moment (current annual increment of volume).
Zeide (2002) had shown algebraically that the current optimal density index,
Sc, is defined as the density index at which current volume growth reaches the
maximum at c. The optimal density was independent of plantation age,
implying that the density, which maximizes the current growth, is the same at
any age level. It was also invariant to diameter, canopy closure, interest rate
flows and merchantable limits (Zeide, 2002). In addition to species and
possibly region, optimal density change with any factor related to density. He
also indicated that insects, diseases, storms and other disturbances are likely
to decrease the optimum because of the damage caused by these factors,
which actually escalates at higher densities.
If at each moment, volume growth is maximal at the highest density, then it
seems that the sum of volume increments (total volume) must also be the
highest. This reasoning is true if volume growth is a function of density only.
Actually, it is a function of two variables, density and diameter. Diameter also
depends on density but not on the current density. As the sum of increments,
diameter has been formed by density that existed in the past. Optimal value of
long-term density is the density that maximizes volume accumulated by the
rotation age and, therefore, its mean annual increment. The results obtained
for loblolly pine by Zeide (2002) showed that the accumulated volume (and its
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mean annual increment) culminates at I = 0.72, which is lower than the
current optimum, I = 1 (where, I indicates relative measure of density = S/Sc,
S = Reineke’s Stand density index, and Sc = Current optimal density index).
This confirmed that, although current optimal density maximizes volume
growth at each moment, it is not optimal in the long run. The same high
density that assures maximal increments at younger ages diminishes diameter
and undermines the total yield.
To improve growth, foresters maintain stand density at a certain fixed level by
regular thinning. In Europe, these levels are known as thinning grades A, B, C
and D. In the United States the levels are defined quantitatively, usually in
terms of residual (after thinning) or average basal area per unit area. As in
Europe, it was found that medium values of basal area are best for volume
growth per unit area. To maximize stand volume, many authors (Chapman,
1953; Wahlenberg, 1960 and Schultz, 1997) recommend reducing basal area in
even-aged loblolly pine (Pinus taeda L.) stands to the residual level of 18 m2/ha
when they reach 27-28 m2/ha. Lately, stand density index has become
popular for specifying the levels. Dean and Chang (2002) recommend growing
loblolly pine between indices to 610 (thinning density) and 390 (residual
density). Williams’ (1994) estimates of these levels are 540 and 390. Similar
values (560 and 390) are used by Doruska and Nolen (1999).
Average optimum density is the simplest but not necessarily the best way to
maximize merchantable volume of teak stands. It may be that the optimum of
current density changes with age. Whereas in the past foresters were
concerned mostly with finding one optimal fixed level of average density, now
the challenge is to find an optimal trajectory of current density. Zeide (2004)
described a method to determine optimal trajectory of stand density and
practical recommendations to implement it. Keeping number of trees per unit
area constant, assures optimal trajectory of stand density. The number
maximizes final yield (or income) because the number provides the lowest
density at the beginning (to increase stem diameter) and full stocking at the
end and as a result, the fastest diameter growth. Such a prescription is called
the minimum number-maximum yield (minimax) strategy. As a result, on good
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sites this strategy reduces expenses on planting and thinning, minimizes root
rot, insect infestation and other possible hazards. At the beginning, minimax
requires only less than 10 per cent of the land for trees. The rest can be used
to diversify land use and grow agricultural crops that do not compete with
trees for light. To remedy shortcomings of minimax, it is suggested to plant
trees in clusters and prune them. Zeide (2004a) had indicated that the real
problem is to find out the best trajectory of density that minimizes the
negative side of density (growth suppression) and capitalizes on its positive
side (increased volume).
2.3. Growth models
A ‘model’ represents a structure showing the proportions and arrangements of
its component parts. Common usage of the term ‘growth model’ encompasses
the mathematical equations, numerical values embedded in those equations,
the logic necessary to link these equations in a meaningful way, and the
computer code required to implement the model in a computer. Growth and
yield models may be precise and realistic, but are limited in their scope and
may require empirical data for calibration. Growth models are of considerable
importance in forest management.
Growth models allow foresters to optimise thinning, fertilization, and virtually
any management activity. Model form and parameters embody the essence of
forestry knowledge and are the basis of contemporary forestry. While there are
no doubts about its significance, there is considerable disagreement on how to
model growth. At present, forest models either merely fit data and predict
future values of measured variables or attempt to understand and describe the
underlying cause-and-effect processes responsible for stand dynamics. The
first class of models is known as empirical models and the second as process-
based models.
2.3.1. Empirical models
The empirical approach to growth modelling is exemplified by equations
selected to maximize fit to the given data set and smooth the data. Their form
and parameters have no ecological or mechanical interpretation. A classic
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example of such models is a polynomial. The Chapman-Richards (Richards,
1959), Gompertz (1825), Korf (Kiviste, 1988), Hossfeld (Peschel, 1938) and
other popular growth equations also belong to this category. Attempts to
design more meaningful growth models represented by the Bertalanffy (1957)
and logistic equations failed because these equations do not fit data well.
They are valued for pragmatic reasons and also for convenience of
calculation. Parameters of these equations are computed to minimize
deviations from data and have no meaning besides serving this practical
purpose. It is believed that empirical models may be useful in practice but
contribute little to our knowledge.
One important development that occurred with respect to the form of growth
model to be used was the analysis of growth equations and was formulated by
Zeide (1993). By analyzing the structure of several growth equations, it was
observed that their diversity is superficial and can be reduced to two basic
differential forms (Zeide, 1993). Each form can be presented as a product of
two modules (expansion module and decline module). In all studied equations
(except the moderately accurate Weibull’s), the expansion module brings the
increment up and describes it as a power function of size. The decline module
pushes the increment down. The decline module has two forms: exponential
and power. This module can be either negative exponential or power functions
of age. This could be explained by a greater number of factors that hinder
growth: scarcity of resources, competition, reproduction, aging, diseases,
herbivory disturbances, etc. The Bertalanffy, Richards, logistic, Gompertz and
other equations belong to the group of Exponential Decline (ED). The Korf,
Hossfeld, Levakovic, and Yoshida equations comprise the group of Power
Decline (PD). The decline module is driven by age, which stands in these
equations as a proxy of all forces that reduce growth.
These two forms of growth equations are
ED: z and PD: ( ) qtpekXtX −=, ( ) qptkXtXz −=, (2.4)
where, k is the scale parameter
p and q are the parameters characterizing the rates of growth
expansion and decline, respectively.
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Zeide (1993) noted that growth of plants results from two opposing factors: the
intrinsic tendency towards unlimited increase (biotic potential) and restraints
imposed by environmental resistance and aging. Expansion tendency prevails
in the beginning of a tree’s life, while growth decline becomes prominent
towards the end. The existing growth equations can be transformed so that the
components that correspond to these two factors are exposed. This
transformation revealed the intrinsic form of the analyzed equations as
or (2.5) tqypky ln ln ln ' ++= qptyky 1' =
or yky = (2.6) qtypky ln ln ' ++= qtpe1'
where, p is the constant of size y or ln y
q is the constant of age t or ln t
p > 0, q < 0, and k1 = ek
k = intercept
In both forms, the expansion component is proportional to ln(y) or is a power
function of size. The forms differ in the way the decline component is
presented. In Equation (2.5) it is proportional to logarithm of age, t. This form
is referred to as the LT-decline or LTD form. The decline component of
Equation (2.6) is directly proportional to age, t. Accordingly, Equation (2.6) is
referred to as TD (T-decline) form. Growth decline of individual trees appears
to be more variable and can be rendered with equal accuracy by a variety of
expressions. Zeide (1993) found that the accuracy of these basic equations
(LTD and TD) is equal when they are applied to individual trees for any tree
variable (diameter, height, volume).
In addition, for individual trees the equation containing tree size as
independent variable was found to be equally successful.
or qyypky ln ln ' ++= qypeyky 1' = (2.7)
where, p = Constant of size y or ln y
q = Constant of age t or ln t
p > 0, q < 0, and k1 = ek
k = Intercept
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Because the decline component is proportional to the size y this form would be
referred to as the Y-decline, or YD form. Zeide (1993) suggested that the best
two variable equations were almost as accurate as equations with three and
four variables. The effect of data type on the accuracy of a given equation
indicated that the effect was especially clear for the YD form; it was the best
for height and diameter growth of individual trees and the worst form for
pooled data for the same variables. The distinguishing feature of the three
equations (LTD, TD and YD) is that growth expansion is proportional to the
logarithm of size. Zeide (1993) discovered a new promising equation form by
combining the three basic forms such as LTD, TD and YD at the expense of
introducing additional parameters.
(2.8) yqptqpqp eykeyktyky 332211321
' ++=
The LTD form corresponds to k2 = k3 = 0, and the TD form arises when k1 = k3
=0. When all three parameters are different from zero, Equation (2.8) becomes
a single general form that includes the discussed forms (LTD, TD and YD) as
special case. However, these equation forms were basically suited for growth
simulation than explicit prediction of yield.
2.3.2. Process-based models
Process-based or simply process models (such as those collected in Dixon et
al., 1990; Korpilahti, 1997; Makela and Landsberg, 2000) aim not only at a
description but also at understanding the underlying cause-and-effect
relationships. Although the promise of process models is yet to be realized,
they are considered as the major achievement of forest science in the twentieth
century.
The bottom-up approach is the key feature of existing process models. These
process models proceed from the particular to the overall result. They start
with describing physiological processes in various tree compartments. Among
these processes are radiation absorption; photosynthesis of shaded and
unshaded conditions; transpiration; respiration of foliage, extent of sapwood
and heartwood, fine and coarse roots, and other tree compartments; like rate
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of senescence; amount of nutrients retranslocated from senescing foliage;
allocation of resources by tree compartments, concentration of nutrients in
each of these, efficiency of carbon conversion; and many others.
The contribution of each process is expressed in the same units, unit of mass.
The bottom-up models present the whole as the sum of its parts. In the words
of the experts, process models "treat plants as consisting of elementary units.
The core of such a model is the description of what happens in a single plant
element. Models can use various elements, such as bud, leaf, internode, stem
segment, etc. A computer program takes care of all the elements and
integrates their activities to the functioning of the whole plant" (Sievanen et
al., 1997). As a result, the structure of these models is simple: the constituent
processes exist on one level and are connected by addition (or subtraction).
Even though bottom-up process models are intended to describe biological
phenomena, their conceptual framework–carbon balance–is adopted from
physics. The ideal of a "process-based forest stand growth model" is borrowed
from physical engineering" (Makela, 1992).
Bottom-up process models are praised for their contribution to the
understanding of key mechanisms of growth and yet they have limited
practical application. The authors of one of the early models write that their
model (with 27 parameters) "provides insights into the relationships between
processes and suggests principles governing growth, but is unlikely to be
useful as a yield predictor; for this purpose a more complex model with more
detailed mechanistic descriptions of the various growth processes would be
required" (McMurtie and Wolf, 1983). Seven years later, Sievanen and Burk
(1990) admitted, "although process-based models have been in widespread
use, their usefulness for forest management has yet to be shown." Still, the
authors are optimistic because "in principle these models have the potential
for being highly applicable in solving various forest management problems."
Yet after seven more years, another carbon balance model (with 34
parameters; several of which are guessed) again "provides mainly a qualitative
description of the growth of an even-aged stand" (Makela, 1997). A more recent
review paper by eight leading modelers published acknowledges that "process-
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based models have not yet been implemented in operational management
systems" (Makela et al., 2000).
Many foresters have tried to realize the promise of the process-based approach
to produce meaningful models of wide applicability. Since 1980, foresters in
New Zealand have made determined efforts in this direction. Several teams of
researchers worked on a comprehensive physiological model that included
various blocks dealing with the interception of radiant energy, photosynthesis,
transpiration, water balance, and other processes. This work has been
summarized by Goulding (1994).
There are several problems with this approach. One of these is that “the
reassembly of these units into a functionally predictive model of a community
seems Utopian. Simulation modelers have abandoned the hope of a realistic
composite model as unworkably complex (Zeide, 2003). Such a model would
be impossible to specify and the massive calculations will inflate error until
the predictions are rendered meaninglessly uncertain, and the purpose of the
model will be obscured by its own complexity” (Zeide, 2003).
The second problem is the incompleteness of bottom-up models. The actual
number of physiological processes is too large to model and some of them are
still unknown. Many important processes are often neglected. Among these
are the adaptation and reproduction. Thus, the index of Dixon’s et al. (1990)
collection of works on forest growth modelling (with 38 contributions) contains
no entry on reproductive effort or seed production. Adaptation ("adaptive
response") is mentioned but not modeled. The number of interactions is still
larger, in fact much larger. And these interactions are critical for both
understanding and prediction (Johnsen et al., 2001).
The third shortcoming is the incapacity of physiological models to utilize tree
measurements and, specifically, the most valuable asset of forest science-long-
term observations on permanent plots. Instead, these models require as input
sophisticated and rarely available information. We do not measure regularly
on our plots radiation absorption, transpiration, rate of senescence, and
annual retranslocation of nutrients. When occasionally we do, the errors are
15
large. Some of these variables cannot be measured at all so that values have
to be guessed. Many problems of parameterization and calibration could be
avoided if we are able to utilize readily available tree and stand measurements.
Given these deficiencies, it is easy to understand why bottom-up process
models, praised for their contribution to knowledge of growth processes, poorly
predict stand dynamics. The current state of forest modelling is a paradox. In
other areas of science, deeper understanding results in more accurate
predictions, wider applicability, and practical usefulness. If knowledge is
power, it is first of all the power of prediction. On the other hand, in forestry
we are told that the principal value of process models is “to improve
knowledge rather than to accurately or precisely predict outcomes” (Korzukhin
et al., 1996). This common view begs a question: do we really need an
“improved knowledge” if its predictions are at variance with reality? Outside
forest modelling, such knowledge is called fallacious rather than improved.
The presumed opposition of understanding and prediction is false. These
concepts are closely related: understanding is a major purpose of science
while prediction is confirmation of its correctness. Only meaningful models
can be consistently accurate. The single reason for model inaccuracy is failure
to understand the studied object. If so, despite their claim to be meaningful,
process models are not. And, conversely, allegedly nonsensical empirical
models may be superior not only in predicting stand dynamics but also in
reflecting growth processes as well.
Shrader-Frechette and McCoy (1993) defend the separation of prediction and
explanation because prediction is a goal of a theory and not a criterion of its
scientific status. They use geology as an example of a strong science, which
can explain earthquakes (by compressional or tensional stresses built up at
the margins of the moving lithospherical plates), but not predict them with
certainty simply because they are not deterministic. Geology may be a strong
science but Shrader-Frechette and McCoy (1993) argument is not. The very
ability to assess the intrinsic unpredictability of earthquakes qualifies as a
precise prediction. Prediction is both a goal of a theory and the main criterion
of its scientific status. Still, there is a difference between the discovery of order
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and the discovery of chaos. Both provide reason to act. The result that the
subject matter of a given science is unpredictable can be considered as a
positive achievement: we can put cross on such a science and switch research
efforts to the areas where our knowledge does make a difference.
It may be that the sterility of bottom-up models is not accidental and
transitory. Perhaps, at this stage of our knowledge the bottom-up approach
alone is not capable of describing growth. Still the listed deficiencies are not
an excuse to revert to empirical data-based models. The advocates of the
bottom-up approach equate it with process modelling: "process-based modelling
can be defined as a procedure by which the behaviour of a system is derived
from a set of functional components and their interactions (Makela et al., 2000).
A complementary, top-down approach was discussed by Landsberg in 1986,
who considered it as a temporary stopgap for physiological bottom-up models.
In his latest book (Landsberg and Gower, 1997), this issue is not mentioned at
all. Actually, the top-down approach is an indispensable component of research
and thinking in general. Any attempt to understand which factors, forces, or
processes have produced the most obvious that we see in a given forest–species
composition, tree size or number–can be deemed as a top-down inquiry.
Biological processes of growth can be divided into the following three linked
groups:
(1) Innate tendency of living beings to grow and multiply; (2) Growth
restraints; (a) Intrinsic processes (aging, growth impediments associated with
increasing size and allocation of resources to reproduction); (b) Extrinsic
processes, chiefly competition stress; and (3) Adaptation to these restraints.
Diving down into the inner mechanisms of tree growth we may learn about the
rate of aging, intensity of competition, increase of proportion of dead tissue,
and infer parameter values that produce a certain tree size. Although this
knowledge can be used to evaluate future growth, by itself those rates, ratios,
and parameters are mainly of academic interest. The top-down approach is
not intended for immediate practical use; rather it is a method to uncover the
forces that produce the measurable variables, identify growth processes, their
hierarchy, and even assess the values of parameters.
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2.3.3. The U-approach in growth modelling
Zeide (2003) proposed a new kind of process models, namely U-approach
model, which combines the merits of both bottom-up and top-down
approaches. The parameters of the model are biologically meaningful and are
directly linked to the physiological and ecological characteristics of the
species. We are interested in deciphering the underlying growth mechanics
only to uncover inner mechanisms of growth in hope of predicting future
diameter, number of trees, volume and other practical variables. The joint, U-
approach is a manifestation of the common strategy of learning known under
many names such as analysis and synthesis, induction and deduction,
differentiation and integration, reductionism and holism. The proposed U-
approach to modelling can be summarized as follows.
1. It uses as input “top” variables, which are easily measured tree and stand
characteristics such as diameter, crown width and stand density.
2. Through these variables at the top-down stage of modelling, we decipher the
underlying “bottom” mechanical, physiological and ecological processes,
starting with multiplicative increase for growth models and tree growth for
mortality models. Then we proceed to uncover more specific processes by
inferring consequences of general processes or splitting them into
constituents. Thus, a growth restraint is a consequence of multiplicative
proliferation. These restraining processes can be divided further into two
component groups: internal and external.
3. All this work is done to produce as output predicted values of the variables
needed for management (bottom-up branch).
Zeide (2003) found that the U-approach promises to be meaningful,
comprehensive, and, because it supplies growth information, practical. Even
though the U-approach also combines two strands of modelling, it does not
sacrifice a theoretical basis for accuracy. Its components reinforce each other
so that the whole is superior to the parts (bottom-up and top-down branches).
Along with uniting these two lines of investigations, the U-approach brings
together both understanding and utility.
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Zeide (2004) had developed certain process-based models and procedures for
projecting diameter and volume growth. For the present study, models
proposed by Zeide (2004) are used with appropriate modifications to suit the
context of the data. The models are based on unified approach (Zeide 2003),
which combines the merits of both top-down and bottom-up enquiry in
scientific investigations. The parameters of the model are biologically
meaningful and are directly linked to the physiological and ecological
characteristics of the species.
2.3.4. Resolution level of growth models
Traditionally, forest growth and yield models are divided into three broad
categories with regard to the structural complexity and input detail viz., (i)
Whole stand models, (ii) Size-class distribution models and (iii) Individual tree
models.
Whole stand models predict the different stand parameters directly from the
concerned regressor variables. The usual parameters of interest are
commercial volume per hectare, crop diameter and crop height. Such models
require stand summary information like age, stand density, site index etc. as
regressor variables. Since age and site index determine the top height,
sometimes only the top height is considered in lieu of age and site index. The
whole stand models can be further grouped according to whether or not stand
density is used as an independent variable in these models. Variable density
whole stand models can assess the effects of variation in stand density (e.g.,
crown cover, basal area) on yield. Traditional normal yield tables do not use
density since the word ‘normal’ implies Nature’s maximum density. Empirical
yield tables assume Nature’s average density. Variable-density models split by
whether current or future volume is directly estimated by the growth functions
or whether stand volume is aggregated from mathematically generated
diameter classes. A second distinction is whether the model predicts growth
directly or uses a two-stage process, which first predicts future stand density
and then uses this information to estimate future stand volume and
subsequently growth by subtraction.
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Models which use information on size distribution of trees (e.g., diameter class
models) are called size-class distribution models. Diameter class models trace
the changes in volume or other characteristics in each diameter class by
calculating growth of the average tree in each class, and multiply this average
by the inventoried number of stems in each class. The volumes are then
aggregated over all classes to obtain stand characteristics.
Individual tree models are the most complex and individually model each tree
on a sample tree list. Most individual tree models calculate a crown
competition index for each tree and use it in determining whether the tree
lives or dies and, if it lives, its growth in terms of diameter, height and crown
size. A distinction between model types is based on how the crown competition
index is calculated. If the calculation is based on the measured or mapped
distance from each subject tree to all trees within its competition zone, then it
is called distance-dependent. Distance-dependent individual tree models
maintain a spatial record of the point density around individual trees. If the
crown competition index is based only on the subject tree characteristics and
the aggregate stand characteristics, then it is a distance-independent model.
Experience to date suggests that distance-dependent tree-level growth models
have proved to be useful tools for simulating various silvicultural practices.
However, other current yield model applications such as inventory projections,
stand validation, and harvest scheduling generally utilize other model types
that provide adequately detailed information at far less cost (Daniels et al.,
1979; Clutter, 1980). Especially, whole stand models combine simplicity and
precision levels that are needed for most of the management purposes.
Moreover, the thesis deals with stand summary features such as the stand site
index, density, and age at inflection point etc. Hence whole stand model
approach was considered as most appropriate for the present study.
2.3.5. Allometric relations
Allometric relations refer to the relationship between parts of an organism, or
in general, that of a biological system. In forestry, allometric relations form the
basis of many prediction equations. For example, allometric relations can be
used for predicting tree or stand attributes, which are difficult to measure, based
20
on certain easily measured characteristics. Typical examples of such prediction
equations are height prediction models based on diameter, volume prediction
models based on diameter and height and most of the growth and yield
prediction models used in forestry. Such models gain importance as modern
scientific forest management relies heavily on well-defined growth and yield
models that can be used to assess the status of forests at any point of time
and also predict the future stand conditions under alternative management
regimes. Traditional prediction models used in forestry are based on ordinary
regression functions. Estimates of parameters in a regression function are
obtained based on the principle of least squares (Montgomery and Peck, 1982).
Estimates obtained through Ordinary Least Squares (OLS) regression method
are unbiased and accurate. OLS produces unbiased estimates of the dependent
variable when the predictor contains no errors. However, this need not be the
case always in the case of allometric relations where both the interdependent
variables may contain errors. Various methods have been developed to relate
variables subject to errors (Bartlett, 1949; Sprent and Dolby, 1980; Ricker,
1984; Leduc, 1987). One of the most popular one is the Reduced Major Axis
(RMA) method. While OLS minimizes the sum of the vertical deviations along
the y-axis, RMA estimates regression parameters by minimizing the sum of the
products of the horizontal (along the x-axis) and vertical deviations. The RMA
slope is the geometric mean of the slope obtained by regressing y on x and the
slope obtained by regressing x on y. An equivalent method to calculate the
RMA slope estimate is to divide the OLS slope by the correlation coefficient
(square root of R2). The variance of the RMA slope is the same as for the
corresponding regression line (Teisser, 1948; Kermack and Haldane, 1950).
Unlike OLS, RMA is designed to estimate parameters when predictors contain
errors. For this reason, RMA is more suitable for this investigation than OLS.
RMA is the maximum-likelihood or least biased estimator of the functional
relationship when theoretical errors of the variables are unknown.
2.4. Intrinsic units in growth modelling
Zeide (2004a) demonstrated the benefits of using intrinsic units of age and size
provided by trees (and other plants) themselves in growth models by choosing
the basic Richards function models (Richards, 1959) as an example. When
21
the age and size at inflection point were employed to rescale the Richards
equation, the model became simpler (reduced number of parameters in the
growth model) and its parameters became more transparent. The age and size
at inflection point are discernible from extraneous observations on trees
(through stem/stump analysis) other than that used for estimating the model
parameters.
The basic Richards equation is:
(2.9) cbteay )1( −−=
where, a, b and c are parameters
It describes the development of any non-diminishing tree dimension y (such as
diameter or height) as a function of a single variable–age, t. This description is
highly artificial because it presumes that all other factors of growth are
constant and the development is a pre-determined unfolding of some innate
design.
The Richards equation belongs to the exponential decline family of growth
equations, which also contains the Bertalanffy, Gompertz, logistic and
monomolecular equations. This family has a common differential form:
y’ = k yp e-qt (2.10)
where, y’ = Derivative of y with respect to age t
k = a1/cbc
p = (c-1)/c
q = b
are parameters (Zeide, 1993).
Equation (2.10) consists of two modules standing for two opposite groups of
growth factors. The expansion module driven by tree size, yp, represents an
innate tendency to grow and multiply, while the decline module, e-qt is a proxy
of growth restraints, both intrinsic (such as aging) and extrinsic (chiefly
competition). The expansion module, yp of Equation (2.10) indicates that
growth is proportional to tree size raised to power of p. This makes clear that
22
because growth is related not only to total mass but also to the living part of it,
or, more specifically, to the absorbing surface.
There are several advantages of presenting the differential Equation (2.10) as
the product of two modules rather as their difference (Zeide, 1993). First, the
multiplicative presentation reveals two basic forms behind existing growth
equations: exponential decline and power decline. When the growth equations
are expressed as difference of two modules, the number of basic forms is twice
as large. Instead of one multiplicative form, there are two for the exponential
decline forms. The second advantage is that the Equation (2.10) is more
convenient for calculations (it can be linearized by taking logs, which cannot
be done for difference). The final advantage is accuracy.
Since Richards (1959) discarded the physiological reasoning employed by
Bertalanffy (1957) to justify the value of parameter p, little is known about
expected values of the parameters and their relationships. Their values are
estimated by fitting the equation to data. It would be helpful if we were able to
assess the parameters from early tree measurements and express them in
terms of tangible landmarks of tree growth such as the inflection point (the
point where increment y’ culminates and the initial period of accelerated
growth shifts to the period of deceleration). This can be done by rescaling the
Richards equation using the age, t$, the tree size, , at inflection point,
instead of the cryptic a, b, and c. The rescaled form of the Richards equation
was reported by Zeide (2004a) and is given as
$y
(2.11) cT−cAY )1( −=
where, , rescaled tree size $/ yyY =
T = / tt , rescaled tree age $
A = Asymptote (maximal tree size in the units of ),which is a
function of c: $y
c
ccA
−=
1
c is a parameter, which is also a function of the values at the
inflection point
23
Parameters of the rescaled equation are palpable, especially from stem
diameter. Zeide (2004a) found that the rescaled form of the model was found
applicable to stands of any site quality and requires no additional parameters
because the intrinsic units serve two purposes: (1) they change with site
quality and (2) they are the equation parameters. Zeide (2004a) found that the
parameter c is site- independent, which was established through mechanical
considerations.
In addition to accuracy, the rescaled Richards equation eliminates the problem
of presenting the parameters as a function of site index. The regular form of
the Richards equation requires an extensive and expensive (in terms of the
number of parameters and computational complexity) modification to reflect
site quality (Goelz and Burk, 1992). The advantages of rescaling are as follows:
1. Clear meaning of parameters and possibility to measure them.
2. Possibility to predict growth from early measurements. Because inflection
occurs at young age, the parameters allow one to predict a large portion of it.
3. Checking validity of growth equations. Comparison of the measured values
of the parameters with their statistical estimates can be used to assess
adequacy of an equation.
4. Interpretation of parameters p and c in terms of crown fractal dimension. It
is shown that parameter p (functionally related to c) is one-third of crown
fractal dimension.
5. Relating the parameters to site quality. Rescaling makes transparent the
relationship between the parameters and site. Parameter c does not change
with site (globally).
6. Using the inflection point as base of site index classification would increase
accuracy of site estimation.
The disadvantage of these units is that, these units are not known prior to
measurement because they are intrinsic to individual trees or stands.
2.5. Fractal geometry and its application in growth modelling
The parameters of the Richards function are associated with eco-physiological
parameters of the species. Biologically, unrestrained growth of trees is very
24
much related to utilization of light and thus related to tolerance and self-
tolerance. Tree crowns, which provide structural support to all photosynthetic
activities of the trees, differ from solid objects of classical geometry. It is
neither a three-dimensional solid nor a two-dimensional photosynthetic
surface. It can be viewed as a collection of holes that serves to conduct
sunlight and gases or as a multilevel hierarchy of clustered dots. The crown is
a hybrid of surface and volume. Their understanding requires new ideas about
spatial relationships. Such objects are called fractals (pigment molecules and
chloroplasts). Fractal geometry provides concepts and tools needed to describe
fractals common in nature (Mandelbrot, 1983). The central concept of this
geometry is fractal dimension. Fractal geometry allows one to condense
information on crown structure into a few meaningful numbers such as fractal
dimension, which is a generalization of the spatial dimension of classical
geometry.
Any spatial dimension, F, be it Euclidean integer dimension or fractal
dimension, is represented by the power of the relationship between the
number of units, N (such as smaller cubes), and the linear size of the unit, r,
which is the length of its side (Mandelbrot, 1983).
FrN 1= (2.12)
Fractal dimension )ln()ln( rNF = (2.13)
More precisely, fractal dimensions reflect two kinds of adaptation to shading:
functional (ability of foliage to function in a wide range of light intensities as in
tolerant species) and structural (permeable crowns with deep penetrating
cavities that let the sunlight in). Both adaptations distinguish the crown
surface from the smooth, two-dimensional surface of Euclidean geometry.
Accordingly, the crown’s fractal dimension, or rather its excess over the
Euclidean dimension, has two components, functional and structural (Zeide,
1990). Only the functional component relates to tolerance.
Many studies report fractal dimensions of two-dimensional projections of
crowns (Morse et al., 1985; Strand, 1990; Gunarsson, 1992; Mizoue and
25
Masutani, 1993). These dimensions, aptly named by Mandelbrot (1983) “ sieve
dimensions”, are different from fractal dimensions of actual crowns occupying
three-dimensional space (called “sponge dimension”). While sponge
dimensions are always greater than 2, sieve dimensions never exceed this
value. Sponge and sieve dimensions are related only by an inequality: their
difference is less than 1 (Pfeifer and Avnir, 1983).
At present, a method for determining fractal dimension of single three-
dimensional crown does not exist. The two-surface method provides such a
dimension for a group of trees (Zeide and Pfeifer, 1991; Corona, 1991; Osawa,
1995). The standard method for determining fractal dimension, the box-
counting method, is not practical. It would require slicing the crown into many
layers without distortion of its structure, subdividing them into cubic boxes,
and counting the number of nonempty boxes. This procedure is repeated
many times using various box sizes. The fractal dimension of the studied
object is one of the parameters of the relationship between the number of
boxes and their size. Technical difficulties make this procedure all but
impossible. While it can easily be applied to obtain a sieve dimension of a
photographed image, dissecting the crown into regular boxes would destroy
the structure one is trying to capture. Besides technical problems, the box-
counting method instills two mental blocks that hamper analysis of three-
dimensional crown geometry. The method uses crown measurements in terms
of regular cubes and consecutive sequences of their sizes and counting.
Zeide (1998) proposed a method, which operates with volume and mass of
natural units of the crown, such as shoots and branches, rather than with
numbers of regular cubes. He pointed out that fractal dimension alone is not
sufficient to describe foliage distribution in the crown because it says nothing
about the density of foliage at a given point. Thus the proposed method makes
it possible to separate purely spatial factors represented by fractal dimensions
from eco-physiological effects characterized by foliage density. Application of
the method showed that neither fractal dimension nor foliage density of the
studied loblolly pines correlates with current diameter increment. At the same
time, there was a pronounced negative correlation between fractal dimension
26
and crown size. These results suggest that as crowns become larger, the
amount of foliage located at the crown periphery increases in proportion to the
foliage amount inside the crown. As a spin-off of this analysis, Zeide (1998)
developed a method for estimating relative foliage density (defined as the ratio
of actual to maximal foliage mass for a given branch).
Fractal dimension can be computed either through parameter estimated from
the growth model or through the digital image processing method or through
box-counting method (Zeide, 2004a). Advances in digital processing make it
possible to measure fractal dimension very easily. Even now fractal dimension
of the crown projection (sieve dimension) could be obtained easily using the
box-counting method (Mandelbrot, 1983). Fractal dimension, F varies between
2 and 3. If the dimension of tree surface F > 2, then ,
where is a parameter of the growth model.
)3/2()3/( >= Fpv
vp
2.6. Past research on thinning and rotation age in teak
Past research on thinning in teak stands have been few, even globally. Perhaps
the first of its kind, Hellinga (1939) made the following observations on natural
thinning in unthinned teak plantations. The initial number of trees per
hectare in plantations of different spacing varied from 2,500 to 5,000. After a
period of 20 years, natural mortality had reduced these numbers to
approximately 1,300 to 1,800 trees per hectare. At the same age, the better
sites tend to grow relatively fewer trees than did poor sites. Yield studies of 21
teak sample plots, which were left unthinned since their establishment 20
years ago, showed that the mean basal area diameter was only 70 to 95 per
cent of that of normally thinned plantations; that the number of trees per ha
was 50 to 250 per cent more, and the total basal area per ha 50 to 100 per
cent more than under normal conditions of thinning. The total tree volume per
ha of the unthinned plots was 20 to 80 per cent more than the total tree
volume of the remaining stand and 5 to 25 per cent less than the total tree
volume of the total stand (remaining stand plus thinnings) of normally thinned
plantations. There was very little difference (only 2 to 10 per cent) in the
weighted mean height of the thinned and unthinned plantations.
27
Khlail (1943) advocated delayed thinning in teak plantations by stating that
heavy early thinnings are inadvisable for the following reasons, viz., (i) Teak
responds well even to late thinnings. (ii) Opening up a young crop will cause
the trees to become branchy. (iii) Pruning would prove too expensive in teak
plantations. (iv) Weed growth can be kept suppressed only as long as the
canopy is closed. (v) The thinnings of the first four years are too small to be
merchantable. (vi) Heavy early opening of the crop may lead to storm damage
among the young shallow-rooted trees. (vii) Drastic opening of the canopy
produces an exposure of site that may result in the conversion of good
productive soil into hard, unproductive laterite.
Venkataramany (1956) gave detailed account of the work and periodic results
in the Wayanad and Nilambur Divisions in Kerala State. Figures were adduced
to show that Craib-type thinnings give rapid diameter increment but less
volume than D-grade thinnings, leaving height unaffected.
Tint and Schneider (1980) reported dynamic growth and yield models for
Burma teak. A computer simulation model in FORTRAN was developed for
analysis of stand basal area and volume growth as functions of diameter class
distribution and site class. Examples of output were presented, consisting of
growth and yield tables for a natural teak selection forest in central Burma
and for teak plantations, giving stand statistics, by 5-year age intervals,
including mensurational and yield data for main crop and thinnings, mean
annual increment and current annual increment. The authors also gave
volume tables used in developing the model.
Abayomi et al. (1985) reported results of analyses of variance of diameter and
height increment of 12 teak thinning trials at 6 sites in Nigeria, comparing 4
treatments: no thinning and thinnings down to residual stockings of 800, 400
and 250 stems per ha. Diameter increment tended to increase with thinning
intensity, while height increment and basal area were less affected by thinning
treatment. It was recommended that teak plantations be thinned at ages 5, 10,
15 and 20 years to residual stockings of 800, 600, 400 and 300 stems
respectively to produce a good stocking of large-sized timber stems by age 50-
60 years.
28
Gonzales (1985) presented growth and yield prediction model for teak (Tectona
grandis Linn.) plantations in the Magat Experimental Forests in Philippines.
Volume equations and tables were developed for merchantable and saw timber
heights from models chosen by stepwise regression with data from Nueva
Vizcaya, Philippines. Merchantable volumes were predicted using an equation
with diameter at breast-height and merchantable height from 0.3 m height to
10- or 20-cm top diameter without bark (respectively for total merchantable
and saw timber heights) as independent variables. More recently, Bermejo et
al. (2004) reported yield tables for teak plantations in Costa Rica based on the
data from permanent sample plots in the region.
Perez (2005) developed a set of intensive management scenarios for teak
plantations in Costa Rica that could lead to alternative timber production
practices with attainable and promising economic returns. The study
consisted of measurement of growth and yield parameters at the stand level
and wood properties at the individual tree level and the interrelationship
between silvicultural management and site conditions. High intensity and on
time thinning yielded both individual tree and stand volume. Pruning up to a
reasonable height and on time caused no reduction in tree growth and stand
yield. Important characteristics, such as heartwood content and wood density,
were found to be related more to tree age than to silvicultural management
practices, especially at early plantation stages. Growth scenario for 20 and 30
year rotations were developed for high, medium and low quality sites. Different
site classes, production objectives, rotation periods, and discount rates
resulted in marked differences in the financial profitability projections of the
developed scenarios.
In India, the existing stocking guides for teak plantations were offered by FRI
(1970). Some recent evidences generated through simulation studies indicated
a need for rethinking in this line (Jayaraman, 1998). The results of the
simulation studies using yield prediction models estimated from a huge set of
temporary sample plots in teak plantations indicated that total yield for a
rotation age of 60 years is enhanced by retaining trees larger in numbers than
that specified by the yield table. Jayaraman and Induchoodan (2004) reported
29
the results of attempts to develop growth simulation models and optimise
stand density for teak plantations in Kerala. The models were similar to that of
Zeide (2004). The present study shall extend the work of Jayaraman and
Induchoodan (2004) using rescaled Richards function based on intrinsic
biological units offered by the trees. The age and size at inflection point of
diameter are proposed to be used for rescaling of the growth model.
Information on inflection point of diameter is obtainable from annual rings of
trees. In forestry, stem analysis is the most dependable practice to generate
historical data on the growth of trees.
2.7. Environmental effects of growing teak
The environmental effects of growing teak plantations are not well documented
so far. However, a few related studies are discussed here.
Although teak is an extremely hardy species capable of growing over a wide
range of edaphic factors, the primary features affecting the growth of the
species with regard to surface and subsurface soils are depth, drainage,
texture, moisture status, and fertility (Seth and Yadav, 1958). Teak is a
pronounced light demander and requires complete overhead light and a fair
amount of space for proper growth and development (Troup, 1921; Tewari,
1992). The environmental effects of growing teak plantations have not been
studied in detail partly because of methodological problems (Thampi, 1997).
In many cases, environmental effects are location specific and are contributed,
either individually or jointly, by a variety of biotic, anthropogenic and
managerial factors. Also, the environmental effects in teak plantation areas are
the net effect of the influence of all interacting factors. Thus, in a long rotation
crop like teak, the changes in environment cannot be monitored at
disaggregated level. However, it has been pointed out that low undergrowth,
increased run-off, soil compaction and erosion in teak plantations have
contributed to the deterioration of soil fertility, which in turn affects the
environment adversely (Shanmuganathan, 1997; Ram and Jana, 1997; Jose
and Koshy, 1972). Decline in soil fertility in successive rotation of teak
plantations in Kerala was noted by Balagopalan and Jose (1982) and
30
Balagopalan and Chacko (2001). It is also a general observation that good
quality teak is restricted to strips along riverbanks and lowering of site quality
occurs at places away from riverbanks.
Effects of growing teak on wider environment are also not well documented.
Few studies have been conducted on carbon absorption and storage ability of
teak plantations. Some of the important ones are reviewed here.
Dabral et al. (1964) studied the effect of dew deposition during two cold
seasons in fully closed stands of Pinus roxburghii, Tectona grandis and
Dendrocalamus strictus. It was observed that dew deposition retarded the
under cover, especially immediately below the crowns. No frost was recorded
inside these plantations. Among these three species least dew deposition was
observed under teak. Hence he opined that plantations (teak) not only helped
in reducing the direct frost injuries to the tender undergrowth, fauna and
innumerable soil microorganisms but also prevented excessive water loss due
to transpiration of the trees, as the dewdrops are still adhering to the canopy.
This according to him again helped in maintaining the soil moisture recharge.
Krishnakumar et al. (1991) studied the influence of rubber and teak
plantations and natural forest on soil properties, nutrient enrichment,
understorey vegetation and biomass recycling at three sites in the Siliguri
subdivision, Darjeeling District of West Bengal. All the three sites had high
input of organic carbon enriching the soil. Teak had the highest organic matter
content in the surface layers. The depletion of organic carbon with depth was
highest for teak and least for natural forest. The results suggested that the
depletion of sub-surface soil moisture would be less under rubber than teak.
The soils under teak showed higher calcium content in the surface layers.
Gogate and Kumar (1993) assessed ecological losses and gains in teak
plantations raised in West Chanda Project Division, Maharashtra vis-à-vis
natural forest areas. The study revealed that clear felling followed by teak
planting would not affect plant diversity in the initial stages of plantation
because during these stages, there was no loss of plant diversity as all the
characteristics of original crop were preserved. The finding was attributed to
31
safety measures adopted at clear felling which involved retention of small
patches of original forest in the form of section and compartment lines,
roadsides, retention of fruit trees etc.
The rates of organic carbon diminution in the soil under different monospecific
tree plantations including teak in Nigeria were studied by Aweto (1995). The
differences between the organic carbon status of their soils and soil under
nearby natural rain forest vegetation were compared. The study indicated that
the tree plantations released more carbon dioxide from the soil into the
atmosphere than the natural forest.
Siringoringo and Gintings (1997) investigated the role of teak in the carbon
sequestration in plantations in Bojonegoro Forest District, East Java. In the
study location, an analysis was undertaken on microclimatic conditions of
light intensity, relative humidity, air pressure and temperature and carbon-di-
oxide in seven stand age classes. Absorption of carbon-di-oxide varied by age
classes of the plantations and absorption ability was highest in old age classes.
Singh (2003) suggested that teak is one of the best suitable tree species in
farm forestry programmes. He also reported that farm forestry holds
tremendous potential for sequestering and storing carbon. Usually, eucalypts,
poplar, teak, kadam (Eucalyptus, Populus, Tectona grandis and Anthocephalus
chinensis, respectively) are being planted in the farm forestry activities.
Reforestation is being considered as a means to capture significant amounts of
carbon, and expected to contribute to soil quality and conservation. If
afforestation is considered as a temporary option to store carbon for several
decades, the amount of carbon removed from the atmosphere over the
decades-long rotation (but prior to harvest) is the same as the standing crop of
carbon at maturity, and the average amount of carbon removed annually is
equivalent to the mean annual growth increment (Schroeder, 1992). He also
indicated that longer the rotation length, the higher the accumulation of
biomass over time, and therefore the greater the mean carbon standing crop.
Rotation length and growth rate interact to determine storage. Growth rate
