20.8.20041

EFIMED Advanced course onMODELLING MEDITERRANEAN FOREST STAND

DYNAMICS FOR FOREST MANAGEMENT

MARC PALAHIHead of EFIMED Office

INDIVIDUAL TREE MODELLING

20.8.20042

Types of empirical growth modelsGreat diversity of models and classifications

Clutter categorized them by the complexity of the

mathematical approach involve• Tabular form

• systems of equations

However, a widely used classification is based on the

modeling unit and output detail;• Whole-stand models

• Individual-tree models

• distance-dependent

• distance-independent

20.8.20043

Why we need individual tree models?

Stand-level models prodive stand level information (G, Hdom, N, V,

etc), and uses stand-level statistics as input data (homogenous stands)

Individual tree models predict the development of each tree within a

forest

- Flexibility to forecast tree growth regardless of species mixture, age

distribution or silvicultural system (any mixture and structure)

- Enable a more detailed description of the stand structure and its

dynamics and more types of treatments can be simulated mixed

and uneven-aged stands can be modeled

- Nowadays possible because of the computing technology

20.8.20044

Designing a growth model

Requires considering: • the resources available; modelling data • the structure of the forest stands, whether they are

even- or uneven-aged or pure or mixed stands • the uses to which it will be for, the input data and

computing technology

20.8.20045

Processes to model

Forest stand development affected by:

GROWTHMORTALITYREGENERATION

MANAGEMENT

20.8.20046

Individual-tree modelsUse individual tree as the basic unit for predicting tree establishment, growth and mortalityConsist of:

-Diameter increment model -Height (increment) model

-Mortality model

-Ingrowth model

Site quality, density, stage, Site index model

Remember the practicability

GROWTH

MORTALITY

REGENERATION

FACTORS

20.8.20047

Modelling dataRepeated observations of dbh and height covering the full range of

expected forest stand situations in site, density, age, management type

(diameter and height growth models)Information of which trees die and how many trees enter the first

diameter class (survival and ingrowth models)

• D_increment = F(tree size, competition, site, age)• Height = F(dbh, site, age)• Survival= F(tree size, competition, age, site)• Ingrowth= F(competition, site)

Computing predictors representing all these variables (tree and stand level)

20.8.20048

Diameter increment models

Explain in detail by Rafa Calama!

Different approaches might be used• Example using linear regression

Ln(Incr) = a + b (tree) + c (comp) + d (site) + є

Predictors representing those factors explaining diameter

growth and providing biologically consistent increment

patterns

Linear and non-linear regression

20.8.20049

Height (increment) models

When remeasured trees for height are available, height increment

models are possible.

-When only static information available, static height models:

H = F (dbh, age, site, ddom)

Non-linear regression

20.8.200410

Ingrowth modelling

Predicting the number of trees entering the first diameter class is

important to make realistic simulations

Ingrowth depends on the species and stand/site conditions

Example:

ING p. nigra= a – b * G + c * Np.nigra/Ntotal + d * ELE – e * ELE2

Linear regression

20.8.200411

Survival modelling (1)Predicting the surviving trees per hectare is a central element of

growth modelling to provide reliable and biologically realistic

simulations of forest stand developmentMortality in natural forests is characterized by long periods of low

mortality and (when there is no management) brief periods of high

mortality, when the self-thinning limit is reachFor a give average tree size there is a limit to the number of tree per

hectare that may co-exist

• Nmax = a * Dg b

The parameters can be obtained by fitting the model plots in the self-

thinning limit

Logistic regression

20.8.200412

Survival modelling (2)

In simulation we assume that mortality is a continuous process

• It can be modelled at the stand level or tree level

• difficult task; big variability 1-10%, many reasons, lack of data

• Remeasurement plots neededTree-level survival models predict for each tree the survival

probability for a given period of time based on tree size, competition

and stand density/site variables.Because the dependent variable is binary (dead or alive – 0 or 1)

• Binary logistic models are commonly used

Logistic regression

eSITETREECOMP

surviveP

3210 ββββexp1

1)(

20.8.200413

Survival modelling (3)

The modelling process:

- test logical predictors (tree, competition, site,) and transformations

- significance and logical signs

- Prediction ability of the model; Chi-square statistic, signal detection

theory

Examples:

Logistic regression

20.8.200414

Survival modelling (4)

From BAL 10 o 20 => 10 times more probable to die (10*0.965)

20.8.200415

Survival modelling (5)

Implementing the models in practice:

- When simulating if each tree dies or survives, usually the estimated probability is

compared to uniformly distributed random number.

- when simulating stand development based on information by diameter classes,

the estimated probability is multiply by the number of trees per hectare in the

diameter class being simulated:

Dbh Classnumber trees Probability

surviving trees

10 300 0,75 225

20 400 0,8 320

30 300 0,9 270

40 70 0,95 66,5

50 15 0,85 12,75

60 3 0,75 2,25

1088 896,5

20.8.200416

Simulation based on tree-level models1. The models are programmed into a simulator

2. Inventory data from the forest stand is needed

3. The simulator reads the data

4. Growth, mortality and ingrowth simulated for half of the period

5. Management interventions are simulated at this point

6. Growth, mortality and ingrowth simulated until the end of period

7. Stand variables, economic parameters, biodiversity indices, etc are

calculated

20.8.200417

Simulation of one time stepIncrement tree ages by the time step

Calculate diameter increment and add to dbh

Calculate new height using a height model

Predict mortality

Predict ingrowth (or regeneration)

Calculate tree volumes

Calculate stand variables

SIgGfBALeAgeddbh

cdbhbaid )ln(1

5

Age

dbhBAL

surviveP

297.2035.0954.3exp1

1)(

TD

dbh

domdom

dom

D

dbhHh

0015.03317.05546.0

3.11.3

20.8.200418

Concluding

Models should be

• simple and easy to use and fulfilling our needs

• biologically consistent according to forest growth laws

and reasonable when extrapolating out of the range of

our data (qualitative evaluation)

• sufficiently accurate (quantitative evaluation)

• flexible to accommodate a range of stand conditions

20.8.200419

ConcludingModels are an abstraction of reality and all of them contain errors BUT they might be very useful to support decision making in forestry

We should always make ourselves some questions when choosing or developing a model;

• Will the model work for my application and input data?• What range of data was/will be used to develop the

model?• Do model assumptions and inferences apply to my

situation? (e.g. type of thinnings, variables, etc)• The end use determines which model/type/approach we

should choose

20.8.200420

Next Rafa Calama;

Diameter increment modelling!

Afterwards;

You will fit diameter increment and survival

models!