Statistics applied to forest modelling
Module 1
Summary

Introduction, objectives and scope
 Definitions/terminology related to forest modelling

Initialization and projection

Allometry in tree and stand variables

Growth functions
 Empirical versus biologically based growth functions
 Simultaneous modelling of growth of several individuals
Expressing the parameters
environmental variables
as
a
function
of
stand
and
Self-referencing functions: algebraic difference and generalized
algebraic difference approach
Definitions/terminology related to
forest modelling
Forest model

A dynamic representation of the forest and its behaviour,
at whatever level of complexity, based on a set of (sub-)
models or modules that together determine the behaviour
of the forest as defined by the values of a set of state
variables as well as the forest responses to changes in the
driving variables
State variables and driving variables

State variables (at stand and/or tree level) characterize
the forest at a given moment and whose evolution in time
is the result (output) of the model:
 Principal variables if they are part of the growth modules
 Derived variables if they are indirectly predicted based on the
values of the principal variables

Driving variables are not part of the forest but influence
its behaviour:
 Environmental variables (e.g. climate, soil)
 Human induced variables/processes (e.g. silvicultural treatments)
 Risks (e.g. pests and diseases, storms, fire)
Modules and components

Model module
 Set of equations and/or procedures that led to the prediction
of the future value of a state variable
 Algorithms that implement driving variables (e.g. silvicultural
treatments, impact of pests and diseases)

Module component
 Equation or procedure that is part of a model module
Modules types

Modules can be briefly classified as
 Initialization modules
 Growth modules
 Prediction modules
 Modules for silvicultural treatments
 Modules for hazards
Forest simulator

Computer tool that, based on a set of forest models and
using pre-defined forest management alternative(s),
makes long term predictions of the status of a forest
under a certain scenario of climate, forest policy or
management

Forest simulators usually predict, at each point in time,
wood and non-wood products from the forest

There are forest simulators for application at different
spatial scales: stand, management area/watershed,
region, country or even continent
Forest simulators at different spatial scales

Stand simulator
 Simulation of a stand

Landscape simulator
 Simulation, on a stand basis, of all the stands included in a
certain well defined region in which the stands are spatially
described in a GIS
 It allows for the testing of the effect of spatial restrictions such
as maximum or minimum harvested areas or maximization of
edges
Forest simulators at different spatial scales

Regional/National simulator – not spatialized
 Simulation of all the stands inside a region, without
individualizing each stand, stands are not connected to a GIS

Regional/National simulator – spatialized through a grid
 Simulation of all the stands inside a region, without
individualization of each stand, stands are not exactly located
but can be placed in relation to a grid
Forest management alternatives and scenarios

Forest management alternative (prescription)
 Sequence of silvicultural operations that are applied to a stand
during the projection period

Scenario
 Conditions (climate, forest policy measures, forest
management alternatives, etc) present during the projection
period
Decision support system

Simulator that includes optimization algorithms that point
out for a solution – list of forest management alternatives
for each stand:
 Multi-criteria decision models
 Artificial neural networks
 Knowledge based systems
Initialization and projection
Initialization and projection

Initialization modules provide the values of state variables
from the driving variables such as
 Site index and/or site characteristics
 Silvicultural decisions (e.g. trees at planting)

Growth modules predict the evolution of the state
variables
The need for initialization modules

When do we need initialization modules?
 When forest inventory does not measure all the state
variables (concept of minimum input)
 In the simulation of new plantations
 For the simulation of regeneration
 In landscape and regional simulators after a clear cut
Compatibilidade entre produção e
crescimento

Embora o crescimento e a produção estejam biológica e
matematicamente relacionados, nem sempre esta relação
foi tida em conta nos estudos de produção florestal

É contudo essencial que os modelos,
construidos, tenham esta propriedade:
ao
serem
 Se eu estimar o crescimento anual em 10 anos e somar os
crescimentos, o valor obtido tem que ser igual àquele que se
obtém se eu estimar directamente a produção aos 10 anos
Compatibilidade entre produção e
crescimento

Sendo Y a produção (crescimento acumulado) e t o tempo
(idade), o crescimento será representado por
dY
 f t 
dt
 A produção acumulada até à idade t será
Y  F t   c ,
onde c é determinado a partir da produção Y0 no instante t0
(condição inicial)
Allometry in tree and stand
variables
As relações alométricas

Diz-se que existe uma relação de alometria linear, ou
relação alométrica, entre dois elementos dimensionais (L
e C) de um indivíduo ou população (no nosso caso,
povoamento florestal), quando a relação entre eles se
pode expressar na forma
L  b Ca

ln L  ln b  a ln C
a constante alométrica, caracteriza o indivíduo num dado
ambiente
b depende das condições iniciais e das unidades de L e C
As relações alométricas

A relação alométrica resulta da hipótese de que, em
indivíduos normais, as taxas relativas de crescimento de L
e C são proporcionais
1 dL
1 dC
a
L dt
C dt
1 dL
1 dC
 L dt  a C dt

ln L  k  a ln C
As relações alométricas

O conhecimento da existência de relações alométricas
entre as componentes de um indivíduo ou povoamento é
bastante importante para a modelação do crescimento de
árvores e povoamentos

É uma das hipóteses biológicas que podemos utilizar na
formulação dos modelos
Growth functions
Growth functions

The selection of functions – growth functions - appropriate to
model tree and stand growth is an essencial stage in the
development of growth models.

Two types of functions have been used to model growth:
 Empirical growth functions
Relationship between the dependent variable – the one we want to
model – and the regressors according to some mathemeatical
function – e.g. linear, parabolic
 Analitical or functional growth functions
Functions that are derived from logical propositions about the
relationship between the variables, usually according to tree growth
principles
 Which should we prefer?
Growth functions

Growth functions must have a shape that is in
accordance with the principles of biological growth:
 The curve is limited by yield 0 at the start (t=0 ou t=t0) and by a
maximum yield at an advanced age (existence of assymptote)
 the relative growth rate (variation of the x variable per unit of
time and unit of x) presents a maximum at a very earcly stage,
decreasing afterwards; in most cases, the maximum occurs very
early so that we can use decreasing functions to model relative
growth rate
 The slope of the curve increases in the initial stage and
decreases after a certain point in time (existence of an inflexion
point)
Schumacher function

The model proposed by Schumacher is based on the
hypothesis that the relative growth rate has a linear
relationshiop with the inverse of time (which means that
it decreases nonlinearly with time):
1 dY
1
k
Y dt
t2

1
1
dY  kd
Y
t
Schumacher function

In integral form:
yA e
k
1
t
A  Y0 e k / t0

where the A parameter is the assymptote and (t0,Y0) is the initial
value

the k parameter is inversely related with the growth rate
Lundqvist-Korf function

Lundqvist-Korf is a generalization of Schumacher
function with the following differential forms:
1 dY
n
k
Y dt
t ( n  1)

1
1
dY  kd
Y
tn
Lundqvist-Korf function

The corresponding integral form is:
YA e
k
1
tn

The A parameter is the assymptote

The k and n parameters are shape parameters:

k is inversely related with the growth rate

n influences the age at which the inflexion point occurs
Lundqvist-Korf function
A - a s s ím p to ta v a riá v e l
B - k v a riá v e l
80
60
70
50
60
50
Y
Y
40
30
40
30
20
20
10
10
0
0
0
10
20
30
40
0
10
20
id a d e
id a d e
90
70
C - n v a riá v e l
30
50
1.00
3.00
5.00
D - a s s ím p to ta e n v a riá v e l
40
20
20
10
10
0
0
0
Lundqvist-Korf function
10
20
30
0
40
10
70
30
40
id a d e
id a d e
90
20
1.00
50
C - n v a riá v e l
3.00
5.00
D - a s s ím p to ta e n v a riá v e l
60
90
80
50
70
40
60
Y
Y
50
30
40
20
30
20
10
10
0
0
0
10
20
30
40
0
10
20
0.50
40
id a d e
id a d e
1.00
30
0.10
7 0 -0 . 4 5
7 0 -0 . 5
9 0 -0 . 4 5
9 0 -0 . 5
Lundqvist-Korf function
n v a riá v e l
4.0
4.0
3.5
3.5
ida de a que ocorre o p.i
ida de a que ocorre o p.i
k v a riá v e l
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0 . 5 0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
1
2
3
4
5
v a l o r d o p a r â m e tr o k
n= 0.1
n= 0.5
6
0.0
0.4
0.8
1.2
v a l o r d o p a r â m e tr o n
n= 1
k = 0.5
k=2
k=5
1.6
Lundqvist-Korf function
n v a riá v e l
20
20
18
18
Y qua ndo ocorre o p.i
Y qua ndo ocorre o p.i
A v a riá v e l
16
14
12
10
8
6
4
16
14
12
10
8
6
4
2
2
0
0
-2 2 0
40
60
80
v a l o r d o p a r â m e tr o A
n= 0.1
n= 0.5
100
0.0
0.4
0.8
1.2
v a l o r d o p a r â m e tr o n
n= 1
A = 90
A = 70
A = 50
1.6
Monomolecular function

Assumes that the absolute growth rate is proportional to
the difference between the maximum yield (asymptote)
and the current yield:
dY
 k A  Y 
dt
Monomolecular function

The corresponding integral form:
k
Y  A 1  c e

t
 Y0
1 
A




ce
k t0



A- assymptote;
k – shape parameter, expressing growth speed
Logistic function

The logistic function is based on the hypothesis that the
relative growth rate is the result of the biotic potential k
reduced by the current yield or size nY (environmental
resistence):
1 dY
 k  nY 
Y dt

Relative growth rate is therefore a decreasing linear
function of the current yield
Gompertz function

This function assumes that the relative growth rate is
inversely proportional to the logarithm of the proportion
of current yield to the maximum yield:
1 dY
Y 
 k log A  log Y   k log  
Y dt
 A

The integral form:
YAe
c e
k t
c  log A  log Y0  e
k t0
Richards function

The absolute growth rate of biomass (or volume) is
modeled as:
 the anabolic rate (construction metabolism)
 proportional to the photossintethicaly active area (expressed as an
allometric relationship with biomass)
 the catabolic ratea (destruction metabolism)
proportional to biomass
taxa anabólica
Anabolic rate
taxa catabólica
Catabolic rate
taxa potencial
de crescimento
Potential
growth rate
taxa de crescimento
Growth rate
c1S=c1 (c0Ym) =c2Ym
c3Y
c2Ym - c3Y
c4 (c2Ym - c3Y),
S – photossintethically active biomass ; Y – biomass; m – alometric coefficient;
c0,c1,c2,c3 – proportionality coefficients; c4 – eficacy coefficient
Richards function

The differential form of the Richards function follows:
dY
 Y m  Y
dt
Richards function

By integration and using the initial condition y(t0)=0, the
integral form of the Richards function is obtained:
Y  A 1  ce

1
 k t 1  m
with parameters m, c, k and A where:


,



1

m

ce
k  1  m 
1
  1  m
A   
 
t0
e
 k t0
Richards function
A - assím ptota variável
B - k variável
80
90
70
80
70
60
60
50
Y
Y
50
40
40
30
30
20
20
10
10
0
0
0
10
20
30
40
0
10
20
idade
90
70
id a d e
0 .0 3
50
C - m variável
80
30
0 .0 5
0 .0 7
D - assím ptota e k variável
80
40
40
30
30
20
20
Richards function
10
10
0
0
0
10
20
30
40
0
10
20
idade
90
70
0 .0 3
50
70
70
60
60
50
50
40
40
Y
Y
80
30
30
20
20
10
10
0
0
20
30
40
0
10
0.2
0 .0 7
20
30
40
id a d e
idade
-0.2
0 .0 5
D - assím ptota e k variável
80
10
40
id a d e
C - m variável
0
30
0.4
7 0 -0 .0 5
7 0 -0 .0 4 5
9 0 -0 .0 5
9 0 -0 .0 4 5
Richards function
k variável
4.0
4.0
3.5
3.5
idade a que ocorre o p.i
idade a que ocorre o p.i
m variável
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.1 -0.5 0.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.1
0.2
0.3
0.4
valo r d o p ar âm etr o m
k=0.1
k=0.3
k=0.55
0.5
0.0
0.2
0.4
0.6
valo r d o p ar âm etr o k
m=0.05
m=0.2
m=0.4
0.8
Richards function
m variável
20
20
18
18
Y quando ocorre o p.i
Y quando ocorre o p.i
A variável
16
14
12
10
8
6
4
16
14
12
10
8
6
4
2
2
0
0
-2 20
40
60
80
valo r d o p ar âm etr o A
m=0.05
m=0.15
100
0.0
0.1
0.2
0.3
valo r d o p ar âm etr o m
m=0.30
A=90
A=70
A=50
0.4
Generalization of Richards
and Lundqvist-Korf functions

Lundqvist function
 Schumacher’s function is a specific case of Lundqvist function
for n=1

Richards function
 Monomolecular, logistic and Gompertz are specific cases of
Richards function dor the m parameter equal to 0, 2, 1
Using growth functions as age-independent
formulations

In many applications age is not known, e.g. in trees that
do not exhibit easy to measure growth rings or in uneven
aged stands

For these cases it is useful to derive formulations of
growth functions in which age is not explicit

The derivation of these formulations is obtained by
expressing t as a function of the variable and the
parameters and substituting it in the growth function
writtem for t+a (Tomé et al. 2006)
Using growth functions as age-independent
formulations

Example with the Lundqvist function
Yt  A e
k
1
1
 k m
t

 lny t A  
tm
k
k
Yt  a  A e
1
t  am
Yt  a  A e
1
1




  k  m a m
  ln y t A  





Families of growth functions
Families of growth functions

The fitting of a growth function to data from a permanent
plot is starightforward
Example:
 Fitting the Lundqvist function to basal area and doiminant
height growth data from a permanent plot
Y A
 1
k 
n
t

e



A - asymptote
k, n – shape parameters
Growth functions
Dominant height
A = 48.75, k = 4.30, n = 0.75
Modelling efficiency = 0.960
50.0
50.0
40.0
40.0
A lt ur a do mina nt e (m)
Á r e a ba s a l (m 2 ha - 1 )
Basal area
A = 58.46, k = 5.13, n = 0.81
Modelling efficiency = 0.995
30.0
20.0
10.0
0.0
30.0
20.0
10.0
0.0
0
5
10
15
20
25
Ida de (a no s )
30
35
40
0
5
10
15
20
25
Ida de (a no s )
30
35
40
But how to model the growth of a series of plots?
This is our objective when developing FG&Y models…
Those plots represent “families” of curves
Using growth functions formulated as
difference equations - ADA

Algebraic difference approach (ADA)
 When formulating a growth function as a difference equation,
it is assumed that the curves belonging to the same “family”
differ just by one parameter - the free parameter
 A growth function with 3 parameters allows for 3 different
formulations, usually denoted by the free parameter
 For example for the Richards function:
Richards-A (model with site specific asymptote)
Richards-k (model with common asymptote)
Ricjards-m (model with common asymptote)
Using growth functions formulated as
difference equations - ADA
Example with the Lundqvist function, formulation with
common asymptote and common n parameter, k as free
parameter (Kundqvist-k):
Y1  A
 1
k 
t n
 1
e




Y2  A
 1
k 
t n
 2
e




Y2  A
 t1 
 
Y1  t 2 
 
n
A
A specific curve of the familty is defined by the value of the free parameter
In practice, the fee parameter is a function of an initial condition (Y0,t0)
Using growth functions formulated as
difference equations - GADA

Generalized algebraic difference approach (GADA)
 One of the problems with ADA is the fact that it originates
formulations that differ just by one parameter
 With GADA it is possible to obtain formulations that have more
than one site-specific parameter
 In GADA parameters are assumed to be function of an
unobservable set of varibales (denoted by X) that expresse site
differences
 The equations is then solved by X, which, for a particular site,
is substituted in the original equation (X0)
Using growth functions formulated as
difference equations - GADA

Example with the Schumacher function
lnY    

t
Suppose that =X and =X, then
lnY   X 
X
t
X
lnY 
1  t
X0 
lnY0 
1  t0
By substituting X0 into the previous expression, we get
t t   
lnY   lnY0  0
tt 0   
Using growth functions formulated as
difference equations - GADA

Another example with the Schumacher function
Suppose now that =X and =X, then
lnY   X 

and
Solving for X:
X


t
tlnY     
t 1
lnY    
X
t
 
X

2 lnY    X       
t 
t

t lnY0     
X0  0
t0  1
Finnally, substituting X0 in the previous expression
lnY    
 t  1t 0 




ln
Y





0
t t 0  1t 
t0 
Expressing parameters as a function of
tree/stand variables

Example with the Lundqvist function fit to basal area
growth of eucalyptus (GLOBULUS 2.1 model) :
G  Ag
n
1  g
kg  
t 
e
n
 G1

 Ag

t1 g1
 t ng 2
2


AgQ Iqe 2
Ag  AgQ Iqe 2
n gi  n g 0  n gQ ln( Iqe )  n gn
k g  k g 0  k gQ Iqe  k gnp
N pl
1000
n gi  n g 0  n gQ ln( Iqe )  n gn
 k gf fe
Ni
1000
Ni
1000
Using mixed-models

Mixed-models (linear and non-linear) “split” the model
error according to different sources of variance, such as:
 Region
 Stand
 Plots
…

When using a model fitted with mixed-models theory it is
possible to calibrate the parameters with random
components by measuring a small sample of individuals

This means that it is possible to use specific parameters
for a particular tree/stand
Which is the best method to model
“families” of growth functions?

There is no best method to model “families” of growth
functions

If appropriate the three methods can be combined in
order to obtain more flexible growth models
FIM !!!