Download Individual-tree models

EFIMED Advanced course on
MODELLING MEDITERRANEAN FOREST STAND
DYNAMICS FOR FOREST MANAGEMENT
INDIVIDUAL TREE MODELLING
MARC PALAHI
Head of EFIMED Office
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20.8.2004
Types of empirical growth models
Great 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
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20.8.2004
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
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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
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20.8.2004
Processes to model
Forest stand development affected by:
GROWTH
MORTALITY
REGENERATION
MANAGEMENT
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20.8.2004
Individual-tree models
Use individual tree as the basic unit for predicting tree establishment,
growth and mortality
Consist of:
-Diameter increment model
GROWTH
-Height (increment) model
MORTALITY
-Mortality model
REGENERATION
-Ingrowth model
FACTORS
Site quality, density, stage,
Site index model
Remember the practicability
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20.8.2004
Modelling data
Repeated 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)
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Linear and non-linear regression
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
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Non-linear regression
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)
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20.8.2004
Ingrowth modelling
Linear regression
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
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20.8.2004
Logistic regression
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 development
Mortality 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 reach
For 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 selfthinning limit
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Logistic regression
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 needed
Tree-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
1
P( survive) 
e
1  exp  β 0  β1 COMP  β 2  TREE  β 3  SITE 
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Logistic regression
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:
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Survival modelling (4)
From BAL 10 o 20 => 10 times more probable to die (10*0.965)
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Survival modelling (5)
Dbh Class
Implementing the models in practice:
number
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
- 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:
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Simulation based on tree-level models
1. 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
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Simulation of one time step
Increment tree ages by the time step
Calculate diameter increment and add to dbh
Calculate new height using a height model
Predict mortality
1
 d  Age  e  BAL  f  ln( G )  g  SI
ingrowth
dbh (or regeneration)
id 5  a  b  dbh  c 
Predict
1
P ( survive) 
 
 dbh   
 3.954  0.035  BAL  2.297  
Calculate1 tree
 
exp  volumes
 

 
 dbh 
Calculate stand variables

h  1.3  H dom  1.3  
 Ddom 
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 Age   



 0.5546 0.3317 dbh   0.0015T

 Dd o m 







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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
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Concluding
Models 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
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20.8.2004
Next Rafa Calama;
Diameter increment modelling!
Afterwards;
You will fit diameter increment and survival
models!
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20.8.2004