1. Statistics applied to forest modelling Module 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 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.  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=t 0 ) 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 (t 0,Y 0 ) 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

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(t 0 )=0, the integral form of the Richards function is obtained: with parameters m, c, k and A where:

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

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

51. 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)

52. 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 In practice, the fee parameter is a function of an initial condition (Y 0,t 0 )

53. 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 (X 0 )

54. Using growth functions formulated as difference equations - GADA  Example with the Schumacher function Suppose that  =X and  =  X, then By substituting X0 into the previous expression, we get

55. Using growth functions formulated as difference equations - GADA  Another example with the Schumacher function Suppose now that  =X and  =X, then and  Solving for X:  Finnally, substituting X0 in the previous expression

56. 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) :

57. 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

58. 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

59. FIM !!!