Statistics applied to forest modelling Module 1

1. Statistics applied to forest modellingModule 1

2. 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 as a function of stand and environmental variables • Self-referencing functions: algebraic difference and generalized algebraic difference approach

3. Definitions/terminology related to forest modelling

4. 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

5. 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)

6. 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

7. Modules types • Modules can be briefly classified as • Initialization modules • Growth modules • Prediction modules • Modules for silvicultural treatments • Modules for hazards

8. 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

9. 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

10. 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

11. 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

12. 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

13. Initialization and projection

14. 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

15. 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

16. 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

17. Compatibilidade entre produção e crescimento • Sendo Y a produção (crescimento acumulado) e t o tempo (idade), o crescimento será representado por • A produção acumulada até à idade t será onde c é determinado a partir da produção Y0 no instante t0 (condição inicial)

18. Allometry in tree and stand variables

19. 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 a constante alométrica, caracteriza o indivíduo num dado ambiente b depende das condições iniciais e das unidades de L e C

20. 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

21. 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

22. Growth functions

23. 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?

24. 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)

25. 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):

26. Schumacher function • In integral form: • where the A parameter is the assymptote and (t0,Y0) is the initial value • the k parameter is inversely related with the growth rate

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

28. Lundqvist-Korf function • 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

29. Lundqvist-Korf function

30. Lundqvist-Korf function

31. Lundqvist-Korf function

32. Lundqvist-Korf function

33. Monomolecular function • Assumes that the absolute growth rate is proportional to the difference between the maximum yield (asymptote) and the current yield:

34. Monomolecular function • The corresponding integral form: A- assymptote; k – shape parameter, expressing growth speed

35. 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): • Relative growth rate is therefore a decreasing linear function of the current yield

36. 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:

37. 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 Anabolic rate Catabolic rate Potential growth rate Growth rate S – photossintethically active biomass; Y – biomass; m – alometric coefficient; c0,c1,c2,c3 – proportionality coefficients; c4 – eficacy coefficient

38. Richards function • The differential form of the Richards function follows:

39. Richards function • By integration and using the initial condition y(t0)=0, the integral form of the Richards function is obtained: with parameters m, c, k and A where:

40. Richards function

41. Richards function

42. Richards function

43. Richards function

44. 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

45. 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)

46. Using growth functions as age-independent formulations • Example with the Lundqvist function

47. Families of growth functions

48. 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 A - asymptote k, n – shape parameters

49. Dominant height A = 48.75, k = 4.30, n = 0.75 Modelling efficiency = 0.960 Basal area A = 58.46, k = 5.13, n = 0.81 Modelling efficiency = 0.995 Growth functions

50. 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