Statistics applied to forest modelling Module 1. Summary Introduction, objectives and scope...

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 individuals0 Expressing the parameters as a function of stand and

environmental variables0 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 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, ao serem construidos, tenham esta propriedade: 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

tfdt

dY

r A produção acumulada até à idade t será

,ctFY

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

CabLCbL a lnlnln

a constante alométrica, caracteriza o indivíduo num dado ambiente

b depende das condições iniciais e das unidades de L e C


A relação alométrica resulta da hipótese de que, em indivíduos normais, as taxas relativas de crescimento de L e C são proporcionais

dt

dC

Ca

dt

dL

L

11

ClnakLlndt

dC

C

1a

dt

dL

L

1


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 functions0 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 functions0 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):

t

1kddY

Y

1

t

1k

dt

dY

Y

12

Schumacher function

In integral form:

t

1k

eAy

where the A parameter is the assymptote and (t0,Y0) is the initial

value

the k parameter is inversely related with the growth rate

0t/k0eYA

Lundqvist-Korf function

Lundqvist-Korf is a generalization of Schumacher function with the following differential forms:

nn tkddY

Yt

nk

dt

dY

Y

111)1(


The corresponding integral form is:

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

nt

1k

eAY


D - assímptota e n variável

0

10

20

30

40

50

60

0 10 20 30 40

idade

Y

70-0.45 70-0.5 90-0.45 90-0.5

B - k variável

0

10

20

30

40

50

60

70

80

0 10 20 30 40

idade

Y

1.00 3.00 5.00

C - n variável

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40

idade

Y

1.00 0.50 0.10

A - assímptota variável

0

10

20

30

40

50

60

0 10 20 30 40

idade

Y

90 70 50


k variável

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 1 2 3 4 5 6

va lor do pa râ m e tro k

ida

de

a q

ue

oc

orr

e o

p.i

n= 0.1 n= 0.5 n= 1

n variável

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.4 0.8 1.2 1.6

va lor do pa râ m e tro n

ida

de

a q

ue

oc

orr

e o

p.i

k= 0.5 k= 2 k= 5


A variável

-2

0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100

va lor do pa râ m e tro A

Y q

ua

nd

o o

co

rre

o p

.i

n=0.1 n=0.5 n=1

n variável

0

2

4

6

8

10

12

14

16

18

20

0.0 0.4 0.8 1.2 1.6

va lor do pa râ m e tro n

Y q

ua

nd

o o

co

rre

o p

.i

A =90 A =70 A =50

Monomolecular function

Assumes that the absolute growth rate is proportional to the difference between the maximum yield (asymptote) and the current yield:

YAkdt

dY

Monomolecular function

The corresponding integral form:

tkec1AY

A

Y1ec 0tk 0

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):

nYkdt

dY

Y

1

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:

The integral form:

A

YlogkYlogAlogk

dt

dY

Y

1

tkec

eAY

0tk

0 eYlogAlogc

Richards function

The absolute growth rate of biomass (or volume) is modeled as: the anabolic rate (construction metabolism)

0 proportional to the photossintethicaly active area (expressed as an allometric relationship with biomass)

the catabolic ratea (destruction metabolism)0 proportional to biomass

Anabolic rateCatabolic ratePotential growth rateGrowth rate

S – photossintethically active biomass ; Y – biomass; m – alometric coefficient;c0,c1,c2,c3 – proportionality coefficients; c4 – eficacy coefficient

taxa anabólica c1S=c1 (c0Ym) =c2Y

m taxa catabólica c3Y taxa potencial de crescimento c2Y

m - c3Y taxa de crescimento c4 (c2Y

m - c3Y),

taxa anabólica c1S=c1 (c0Ym) =c2Y

m taxa catabólica c3Y taxa potencial de crescimento c2Y

m - c3Y taxa de crescimento c4 (c2Y

m - c3Y),

Richards function

The differential form of the Richards function follows:

YYdt

dY m

Richards function

By integration and using the initial condition y(t0)=0, the

integral form of the Richards function is obtained:

,ce1AY m1

1tk

with parameters m, c, k and A where:

m1k

eec 00 tk

tm1

m1

1

A

Richards function

D - assímptota e k variável

0

10

20

30

40

50

60

70

80

0 10 20 30 40

idade

Y

70-0.05 70-0.045 90-0.05 90-0.045

B - k variável

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40

idade

Y

0.03 0.05 0.07

C - m variável

0

10

20

30

40

50

60

70

80

0 10 20 30 40

idade

Y

-0.2 0.2 0.4

A - assímptota variável

0

10

20

30

40

50

60

70

80

0 10 20 30 40

idade

Y

90 70 50

Richards function

m variável

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-0.1 0.0 0.1 0.2 0.3 0.4 0.5

valor do parâmetro m

ida

de

a q

ue

oc

orr

e o

p.i

k=0.1 k=0.3 k=0.55

k variável

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.2 0.4 0.6 0.8

valor do parâmetro k

ida

de

a q

ue

oc

orr

e o

p.i

m=0.05 m=0.2 m=0.4

Richards function

A variável

-2

0

2

4

6

8

10

12

14

16

18

20

20 40 60 80 100

valor do parâmetro A

Y q

ua

nd

o o

co

rre

o p

.i

m=0.05 m=0.15 m=0.30

m variável

0

2

4

6

8

10

12

14

16

18

20

0.0 0.1 0.2 0.3 0.4

valor do parâmetro m

Y q

ua

nd

o o

co

rre

o p

.i

A=90 A=70 A=50

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

mt

1k

t eAY

m1

t Ayln

kt

mat

1k

at eAY

mam

1

Atyln

k

1k

at eAY

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

nt

keAY

1A - asymptotek, n – shape parameters

Growth functions

0.0

10.0

20.0

30.0

40.0

50.0

0 5 10 15 20 25 30 35 40

Idade (anos)

Áre

a b

asa

l (m

2ha

-1)

0.0

10.0

20.0

30.0

40.0

50.0

0 5 10 15 20 25 30 35 40

Idade (anos)

Altura

dom

inante

(m

)

Basal areaA = 58.46, k = 5.13, n = 0.81Modelling efficiency = 0.995

Dominant heightA = 48.75, k = 4.30, n = 0.75Modelling efficiency = 0.960

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):

A specific curve of the familty is defined by the value of the free parameter

n

tt

AY

AY

2

1

12

In practice, the fee parameter is a function of an initial condition (Y0,t0)

nt

k

eAY 1

1

1

nt

k

eAY 2

1

2

nt

k

eAY 1

1

1

nt

k

eAY 2

1

2

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)


Example with the Schumacher function

Suppose that =X and =X, then

By substituting X0 into the previous expression, we get

t

Yln

t1

YlnX

t

XXYln

0

00 t1

YlnX

0

00 tt

ttYlnYln


Another example with the Schumacher function

Suppose now that =X and =X, then

and

Solving for X:

Finnally, substituting X0 in the previous expression

t

XYln

1t

YlntX

0

00

0

tYln

t1t

t1t

tYln

t

XYln

t

X

tXYln2

1t

YlntX

0

000

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) :

1000

Nn)Iqeln(nnnIqeAA

A

GAG i

gngQ0ggi2

gQgt

t

g

1g2

2gn2

1gn1

gn

g t

1k

g eAG

fek1000

NkIqekkk gf

plgnpgQ0gg

1000

Nn)Iqeln(nnnIqeAA

A

GAG i

gngQ0ggi2

gQgt

t

g

1g2

2gn2

1gn1

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 !!!

Statistics applied to forest modelling Module 1. Summary Introduction, objectives and scope...

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