EFIMED Advanced course on MODELLING MEDITERRANEAN FOREST STAND DYNAMICS FOR FOREST MANAGEMENT

1. EFIMED Advanced course on MODELLING MEDITERRANEAN FOREST STAND DYNAMICS FOR FOREST MANAGEMENT INDIVIDUAL TREE MODELLING MARC PALAHI Head of EFIMED Office

2. 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

3. 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

4. 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

5. Processes to model • Forest stand development affected by: • GROWTH • MORTALITY • REGENERATION • MANAGEMENT

6. Individual-tree models • Use individual tree as the basic unit for predicting tree establishment, growth and mortality • Consist 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

7. 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)

8. 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

9. 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)

10. Linear regression 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

11. 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 self-thinning limit

12. 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

13. 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:

14. Survival modelling (4) From BAL 10 o 20 => 10 times more probable to die (10*0.965)

15. 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:

16. 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

17. 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 • Predict ingrowth (or regeneration) • Calculate tree volumes • Calculate stand variables

18. 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

19. 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

20. Next Rafa Calama; • Diameter increment modelling! • Afterwards; • You will fit diameter increment and survival models!