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Dynamical implications of the variability representation in site-index modelling

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ORIGINAL PAPER
Dynamical implications of the variability representationin site-index modelling
Oscar Garcı´a
Received: 16 April 2010/Revised: 12 August 2010/Accepted: 23 November 2010/Published online: 21 December 2010
The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract
Issues in the development and formulation of forest site-index models are examined, linking the forestryterminology and methods to standard mathematical con-cepts. Variability complicates interpretation. Three sourcesof variation are distinguished: between sites, within sites,and observation error, with the article focusing mainly onthe second one. Two site-index deﬁnitions arising fromdifferent views about the variability are contrasted. Mod-elling based on algebraic difference equations (ADE’s) isanalyzed in detail, relating it to concepts of state spaceﬂows used in modern dynamical systems theory. It isshown that, given a stand current state, ADE’s predictgrowth rates that are independent of site quality.
Keywords
Forest growth and yield
Site productivity
Algebraic difference equations
ADA
GADA
Differential equations
Dynamical systems
Introduction
Site-index models relate height, age, and site quality(potential productivity) in even-aged single-species stands.They are used for predicting stand height development andfor assessing site quality. The principles can be traced back to the 18th Century (Batho and Garcı´a 2006), and variousapproaches are described in books such as Belyea (1931);Spurr (1952); Assmann (1970); Clutter et al. (1983); von
Gadow and Hui (1999); Pretzsch (2009). General reviews
have been published by Jones (1969); Carmean (1975);
Ha¨gglund (1981); Ortega and Montero (1988); Grey
(1989). Some of the details are subtle. Assumptions andinterpretations are not always clear and explicit, leading tomisunderstandings and controversy. We focus here onsome implications of natural variability of stand develop-ment. Mathematical and modelling aspects are stressed, butdetailed statistical procedures are beyond the scope of thisarticle.Three sources of variability may be identiﬁed: (a)between sites, (b) within sites, and (c) observation error.These are illustrated in Fig. 1. The between-sites variationgives rise to a family of site curves, each representing aheight–age trajectory for a certain site quality. It is dis-cussed brieﬂy in the next Section. For a given site, indi-vidual stands will deviate from the nominal site curve dueto weather ﬂuctuation and other factors. The implicationsof this within-site variability are the main topic here. Inaddition, height observations (and sometimes age) aresubject to sampling and measurement error. These errorscan be important in devising appropriate estimation andstatistical inference procedures, but will not be discussed indetail.Over time, site-index modelling has developed its ownmethods and terminology. The article makes an effort tolink these to standard mathematical concepts. It is hopedthat tapping into a wider pool of knowledge may facilitatefuture progress.
Variation between sites
Differences in height growth across sites are the basis of the site-index concept. Ignoring some of the natural
Communicated by G. Ka¨ndler.O. Garcı´a (
&
)University of Northern British Columbia, 3333 University Way,Prince George, BC V2N 4Z9, Canadae-mail: garcia@unbc.ca
1 3
Eur J Forest Res (2011) 130:671–675DOI 10.1007/s10342-010-0458-0
variation, the main ideas are not difﬁcult to understand. It isassumed that stands follow height–age trajectories char-acteristic of each site quality, the site-index curves (or
sitecurves
) drawn with dots in Fig. 1. The curves do notintersect, except possibly at the srcin, and greater heightsat any particular age indicate higher site quality.Equation-based models usually start with a growth curvefunction
H
=
g
(
A
), where
A
is stand age and
H
is somemeasure of stand height. The growth curve is made to varywith site quality by including a site-dependent parameter,say
q
, so that
H
=
g
q
(
A
)
=
g
(
A
,
q
). For instance, with theSchumacher (1939) function
H
¼
a
exp
ð
b
=
A
Þ
, typicallyone of the parameters
a
or
b
is taken as site-dependent (or
local
), while the other is assumed to be common to all sitesand stands (
global
). Thus, one may have a model where thesite curves differ by an
H
-scale factor,
H
¼
q
exp
ð
b
=
A
Þ
(called
anamorphic
), or by an
A
-scale factor,
H
¼
a
exp
ð
q
=
A
Þ
. More generally, both original parametersmight be assumed to be site-dependent, as in
H
¼
ab
q
exp
ð
q
=
A
Þ
, where
q
is local, and
a
and
b
are newglobal parameters.The local parameter
q
serves only as a label for theindividual site curves and can be chosen in different ways.Curves may be labeled simply by discrete quality classes,often with roman numerals. The most common continuouslabeling scheme uses a
site index
,
S
, deﬁned as the curveheight at some reference
base age A
b
. The site index isrelated to any other site-dependent parameter
q
through
S
¼
g
q
ð
A
b
Þ
.
Variation within sites
Clearly, real stands will inevitably deviate from the curvespeciﬁed by any deterministic model. The model does notnecessarily ignore this and the curve may be interpreted asa point estimate, a predicted, expected, or most likelyheight–age trajectory. We shall not be speciﬁc about thedifferences among these (mean, mode, median, etc.) andwill say ‘‘predicted’’ or ‘‘nominal’’.Which site index?A ﬁrst consequence of this within-site variability is theexistence of different deﬁnitions of site index. Someauthors have explicitly or implicitly deﬁned site index asthe actual height reached by a particular stand at the baseage. This is a property of the stand and is different from thedeﬁnition based on predicted height given in the previousSection, which is a property of the site. ‘‘Stand site index’’corresponds to point A in Fig. 1 and ‘‘site site index’’ topoint B.Deﬁnitions cannot be right or wrong, but the properstatistical treatment differs, and lack of clarity on this pointcan (and has) lead to misunderstanding and controversy.Under the
stand site index
view, models appropriate forpredicting height may differ from those for assessing sitequality, and statistical analysis typically involves error-in-variables situations (Curtis et al. 1974; Goelz and Burk 1992). The
site site index
approach is more abstract,although it may be closer to the srcinal idea.Focusing instead on local and global parameters side-steps these issues.DynamicsThe nominal model height–age trajectories correspond toa prediction ‘‘at birth’’. By analogy to a person’s lifeexpectancy, and for similar reasons, a future height pre-dicted at birth can be expected to be different from thatpredicted later in life. Generally, for a stand growing in sitequality
q
, one knows (or has an estimate of) its height
H
1
atage
A
1
and wants to predict the height
H
2
at some other age
A
2
. Writing
t
¼
A
2
A
1
for the prediction interval, thepredicted height is some function
H
2
¼
F
q
ð
A
1
;
H
1
;
t
Þ
:
ð
1
Þ
It is assumed that this function is continuous and smooth(differentiable). These assumptions are implicit in the useof growth functions like those in Sect. 2, even though themodel does not usually pretend to reproduce the seasonalgrowth ﬂuctuations within a year (but see Garcı´a 1979,1999, for one way of doing this).The function (1) is special in that it must satisfy twoconsistency conditions: prediction over a zero-length inter-val must return the starting value, i.e.
F
q
ð
A
1
;
H
1
;
0
Þ ¼
H
1
,andtwo predictionsover consecutive intervalsmustgive the
Fig. 1
Nominal site-index curve (
dashed
), actual trajectory (
solid
),and observations (
isolated dots
).
A
and
B
indicate different site-indexdeﬁnitions (see text)672 Eur J Forest Res (2011) 130:671–675
1 3
same result as a single prediction over the whole:
F
q
½
A
1
þ
t
;
F
q
ð
A
1
;
H
1
;
t
Þ
;
u
¼
F
q
ð
A
1
;
H
1
;
t
þ
u
Þ
(Sullivan and Clutter1972; Clutter et al. 1983, p. 123; Garcı
´a 1979, 1994)
1
. Sucha function, or more precisely (1) together with the obviousage ‘‘prediction’’ function
A
2
¼
A
1
þ
t
, is known as a (glo-bal)
transition function
(Padulo and Arbib 1974; Garcı´a1994, and references therein) or a
ﬂow
(Arnold 1973, Chap.1) (a semi-ﬂow if predictions back in time,
t
\
0, are notallowed). In forestry, these have been called also differenceequations (Clutter et al. 1983), algebraic difference equa-tions (ADE, Cieszewski and Bailey 2000), or self-refer-encing functions (Northway 1985), at least whenindependent of
q
(see below).Any ﬂow (1) can be obtained as the solution of anordinary differential equation (ODE)d
H
d
t
¼
f
q
ð
A
;
H
Þ ð
2
Þ
(and d
A
=
d
t
¼
1), which is sometimes more convenient(Arnold 1973). In fact, modelling often starts with an ODEformulation, and the transition function arises throughintegration. The reversal of this classical view by theRussian school of Anosov, Arnold, and others, startingfrom ﬂows as the more primitive concept, is attractivebecause it is often argued that ODE’s are not appropriatefor forest modelling. It is also closely related to the ADEideas introduced by Clutter (1963) and Bailey and Clutter(1974).Alternatively, ﬂows may be described by an
invariant
,or ﬁrst integral, an expression that remains constant overtrajectories of the ﬂow. A trajectory is the curve generatedby varying
t
in (1), for ﬁxed
A
1
and
H
1
. This relates also toADE’s, as shown below. For multivariate generalizations,see Garcı´a (2010).
Site curves and ADE’sThe algebraic difference approach (ADA/GADA)
2
pro-duces a ﬂow or ADE compatible with a given site equation
H
¼
g
ð
A
;
q
Þ
(Bailey and Clutter 1974; Clutter et al. 1983;
Cieszewski and Bailey 2000). The usual procedure consistsof solving for the local parameter,
q
¼
u
ð
A
;
H
Þ
, andequating the value at two height–age points:
u
ð
A
1
;
H
1
Þ ¼
u
ð
A
2
;
H
2
Þ
:
Solving for
H
2
, one obtains the ﬂow equation (ADE).For example, with the anamorphic Schumacher model
H
¼
q
exp
ð
b
=
A
Þ
;
q
¼
H
exp
ð
b
=
A
Þ
;
H
1
exp
ð
b
=
A
1
Þ ¼
H
2
exp
ð
b
=
A
2
Þ
;
and
H
2
¼
H
1
exp
ð
b
=
A
1
Þ
=
exp
ð
b
=
A
2
Þ ¼
H
1
exp
½
b
ð
1
=
A
1
1
=
A
2
Þ
.Note that
u
ð
A
;
H
Þ
is an invariant, constant for all thepoints on a trajectory. Alternatively, one may differentiatethe invariant and obtain the ODE for the ﬂow:d
H
exp
ð
b
=
A
Þ
d
A
¼
0d
H
d
A
exp
ð
b
=
A
Þ þ
H
exp
ð
b
=
A
Þð
b
=
A
2
Þ ¼
0
;
from where,d
H
d
A
¼
b H A
2
:
This ODE was actually the starting point of Schumacher(1939).The ADE is a relationship between any two points lyingon the same site curve. The corresponding ODE is calledthe ODE of a one-parameter family of curves (e.g. Agnew1960, Chap. 4); in this case, the family of site curvesparameterized by
q
.The ADE, or its ODE, predicts that any stand of sitequality
q
that currently lies on the site-
q
curve will con-tinue to follow that curve, as one might expect. However,that is not the only ﬂow and ODE with this property. Onecould use any other invariant in the derivation, not just theone that corresponds to the local parameter. A few possi-bilities for the anamorphic Schumacher are shown inTable 1. The ADE, ﬁrst row, is the (only) ﬂow that doesnot depend on site quality. The one on the second row isobtained through solving for
b
; the others cannot beobtained by the method of equating parameters. The ODEin row 3 is a function of
A
, and that in 4 is a function of
H
. There is an inﬁnity of ODE’s depending on both
A
and
H
(and
q
) that produce the same growth curve, startingfrom a point on the curve. In other words, a site equation
H
=
g
(
A
,
q
) is not sufﬁcient by itself to predict futuredynamics when the stand deviates from the nominal curve;hypotheses about the growth rate are needed.PredictionConsider a stand for which we know, or have estimates of,the current age
A
1
, height
H
1
, and the site quality
q
. In theabsence of within-site variability,
ð
A
1
;
H
1
Þ
would be on thesite-
q
curve and, as just explained, any of the ﬂows orODE’s discussed in the previous section then give the samepredictions, following the nominal site curve. In reality, thestand would almost certainly have deviated from the
1
In forestry, the property has been called
path invariance
. Mathe-matically, this is a one-parameter (
t
) continuous group of transfor-mations. An example of Lie group, named after the Norwegianmathematician Sophus Lie (pronounced ‘‘Lee’’).
2
This terminology should not be confused with the standardmathematical meanings. In mathematics, difference equations dealwith sequences of uniformly spaced values. An algebraic equationmay contain elementary operations and rational exponents, excludingexponentials and other transcendental functions (James, 1992).Eur J Forest Res (2011) 130:671–675 673
1 3
nominal curve, and then the different ﬂow and ODEequations produce different predictions.This is illustrated in Fig. 2, for an anamorphic Schum-acher site model with
b
¼
18. Representative site curvesare drawn with dashes and labeled with the site index forbase age 25. The Clutter-Bailey ADE always follows thesite curve passing through the current point, regardless of site quality. On the other hand, the solid curves show thetrajectories predicted by the ﬂow on the forth row of Table 1 for stands of site index 20 that have deviated aboveor below the nominal curve.ADE’s describe site curves well and, by eliminatinglocal parameters, make it possible to ﬁt site models usingstandard nonlinear regression packages. But their repre-sentation of growth dynamics in the presence of within-sitevariation is questionable. A site-index 20 stand that hap-pens to be on the site-index 18 curve can be expected togrow faster than a site-index 18 stand that so far has stayedthe course.Among the alternatives, there are good biological andother reasons to prefer the one where the growth rate (2)depends on size (
H
), but not on age (e.g. Pen˜uelas 2005).For the Schumacher, it is the one on row 4 of Table 1 andin Fig. 2.Figure 3 displays similar predicted trajectories from themodel of Hu and Garcı´a (2010). It is based on an age-
invariant Bertalanffy-Richards ODE. For another example,see the
EasySDE User Guide
at http://forestgrowth.unbc.ca/sde.One situation where the various predictions coincide iswhere the only information available about a stand is thecurrent age and height, without any knowledge of sitequality or previous measurements. Then
q
must be inferredfrom the starting point, and it can be seen that the resultingtrajectories will be the same. This might be a commonoccurrence when applying these systems in practice. Duringmodel development, however, there are usually multiplemeasurementsinasampleplot,andADAessentiallyderivessite quality separately for each pair of measurements,ignoring the restriction of a common value for the plot. Thequestion of if discarding some information is compensatedby avoiding the complications of having to deal with localparameters would depend on the data and other consider-ations, and seems difﬁcult to answer in general.
Table 1
Some ﬂows and ODES that generate
H
¼
q
exp
ð
b
=
A
Þ
Invariant Flow equation ODE1
H
exp
ð
b
=
A
Þ
H
2
¼
H
1
exp
ð
b
=
A
1
b
=
A
2
Þ
d
H
=
d
A
¼
bH
=
A
2
2
A
ln
ð
H
=
q
Þ
H
2
¼
q
ð
H
1
=
q
Þ
A
1
=
A
2
d
H
=
d
A
¼
H
ln
ð
q
=
H
Þ
=
A
3
H
q
exp
ð
b
=
A
Þ
H
2
¼
H
1
þ
q
½
exp
ð
b
=
A
2
Þ
exp
ð
b
=
A
1
Þ
d
H
=
d
A
¼
bq
exp
ð
b
=
A
Þ
=
A
2
4
A
þ
b
=
ln
ð
H
=
q
Þ
H
2
¼
q
exp
½
b
=
ð
A
1
A
2
þ
b
=
ln
ð
H
1
=
q
ÞÞ
d
H
=
d
A
¼
H
ln
ð
H
=
q
Þ
2
=
b
5
q
=
H
exp
ð
b
=
A
Þ
H
2
¼
q
=
½
exp
ð
b
=
A
2
Þ
exp
ð
b
=
A
1
Þ þ
q
=
H
1
d
H
=
d
A
¼ ð
b
=
q
Þ
H
2
exp
ð
b
=
A
Þ
=
A
2
Fig. 2
Dashed
: Site-index curves and ADE trajectories, anamorphicSchumacher model with
b
¼
18.
Continuous
: predicted trajectoriesfor site index 20 according to the fourth row of Table 1. Site indicesfor base age 25
Fig. 3
From model by Hu and Garcı´a (2010).
Dashed
: site-indexcurves for sites 19, 20, 21.
Solid
: projected heights for stands growingin site quality 20, currently at the dot locations. ADE predictionsfollow the
dashed curves
674 Eur J Forest Res (2011) 130:671–675
1 3
Conclusions
The site modelling methods pioneered by Clutter and othersdeviated from the classical ODE-centered approaches todynamical systems common in other ﬁelds. Although theymight have seemed ad hoc, it is remarkable how theyactually paralleled to some extent more recent develop-ments in Mathematics (Arnold 1973; Anosov et al. 1997).
Given current stand conditions, the growth rate in ADE/ ADA/GADA models is independent of site quality. Thismight be seen as a conceptual ﬂaw. Deviations from thenominal curves, however, are unlikely to be large, so thepractical implications are not entirely clear. Certainly,ADA made feasible the development of good site models ata time when computing resources were limited. The com-putational and ﬂexibility advantages of ADA/GADAtechniques might well carry into the future.Only essentially deterministic aspects of growth fore-casting have been discussed, dealing with predicted ornominal trajectories. A superimposed stochastic structure isimportant for hypothesis testing and in the search for goodestimators. One natural extension is to include environ-mental perturbations in the ODE’s, possibly adding alsoobservation errors (Hotelling 1927; Seber and Wild 2003,
Sect. 7.5). It is tempting to use hierarchical modelling forthe sources of variation, treating the local parameter as‘‘random’’ (Snijders 2003; Hall and Bailey 2001); but it
should be remembered that height–age data is rarely arandom sample from the target population, and the effectsof violating this assumption are unclear.
Acknowledgments
This work owes much to extensive discussionsover the years with Keith Rennolls, and with Chris Cieszewski.Encouragement and suggestions from Mike Strub are gratefullyacknowledged.
Open Access
This article is distributed under the terms of theCreative Commons Attribution Noncommercial License which per-mits any noncommercial use, distribution, and reproduction in anymedium, provided the srcinal author(s) and source are credited.
References
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