Improving the accuracy of predicted diameter and height distributions

1. Improving the accuracy of predicted diameter and height distributions Jouni Siipilehto Finnish Forest Research Institute, Vantaa E-mail: jouni.siipilehto@metla.fi

2. Diameter distributions are needed in Finnish forest management planning (FMP) individual tree growth models FMPinventory systemcollect tree species-specific data of the growing stock within stand compartments Stand characteristics consists of: basal area-weighted dgM, hgM age (T) and basal area (G) Number of stems (N) is additional character, which is not required Introduction

3. The objective of this study: to examine whether the accuracy of predicted basal-area diameter distributions (DDG) could be improved by using stem number (N) together with basal area (G) in terms of degree of determination (r2) in terms of stem volume (V) and total stem number (N), when G is unbiased Objectives

4. Study material consisted of: 91 stands of Scots pine (Pinus sylvestris L.) 60 stands of Norway spruce (Piceaabies Karst.) both with birch (Petulapendula Roth. and P. pubescent Ehrh.) admixtures in southern Finland about 90–120 trees/stand plot dbh and h of all trees were measured Test data consisted of NFI-based permanent sample plots in southern Finland 136 for pine 128 for spruce about 120 trees/cluster of three stand plots Study material

5. The three-parameter Johnson’s SB distribution bounded system includes the minimum and the maximum endpoints the minimum of the SB distribution (x) was fixed at 0 fittedusing the ML method to describe the basal-area diameter distribution (DDG ) transformed to stem frequency distribution (DDN) Diameter distribution

6. Johnson’s SB distribution is based on transformation to standard normality in which -z is standard normally distributed variate g and d are shape parameters x and lare the location and range parameters d is diameter observed in a stand plot Distribution function

7. Species-specific models for predicting the SB distribution parametersd and l Linear regression analysis The models were based on either predictors that are consistent with current FMP (ModelG) or those with the addition of a stem number (N) observation (ModelGN) Predicting the distribution

8. When predicting the SB distribution, parameter g was solved according to known d and land median dgMusing Formula Thus, known median was set for predicted distribution. ”Percentile method”

9. Single stand variables: dgM, G,NorT did not correlate closely with the shape parameter d of the SB distribution In ModelGN, stand characteristics were linked together for ”shape index” y in which ”Shape index”

10. The behaviour of the shape indexψ Stem frequency (solid line) and basal area distributions (dotted line)

11. Correlation between parameter dand shape index yfor spruce and pine • Correlation r = 0.57 and 0.68 for pine and spruce, respectively

12. ModelG dgM and T explained l, and stem form (dgM/hgM) was the additional variable explaining d r2for l and d 0.22 and 0.05 for pine 0.40 and 0.28 for spruce ModelGN Shape index yalone or with dgM explained land d r2for l and d 0.28 and 0.38for pine 0.37 and 0.50for spruce Results: Prediction models

13. The relative bias and the error deviation (sb) of the volume and stem number in the test data

14. The predicted DDGs (above) and the derived DDNs for spruce and pine, when y=1.0, 0.77 and 0.63

15. ModelGN is capable of describing great variation in N within fixed dgM and G Example dgM=20 cm, G=20 m2ha-1 if y = 1.00 then N = 705 and 790 ha-1 if y = 0.63 then N = 1020 and 1100 ha-1 for pine and spruce, respectively UnbiasedN = 640 and 1020 ha-1 Advantages

16. Height distribution is not modelled for FMP purposes It is produced with a combination of dbh distribution and height curve models only expected valueof height is used for each dbh class height distribution has become of great interest latelyfrom stand diversitypoint of view available feeding, mating and nesting sites for canopy-dwelling organisms Objective to examine how the goodness of fit in marginal height distributions can be improved using the within dbh-class height variation in models Height distribution

17. Näslund’s height curve Linearized form for fitting power a =2 and 3 for pine and spruce respectively b0 and b1 estimated parameters Residual error e: homogenous variance normally distributed Height model including error structure

18. The residual variation (sez) of efrom linearized model transformation to concern real within-dbh-class height variation (seh) using Taylor’s series expansion Error structure handling

19. Error structure behaviour • funtion of diameter and height • dependent on height curve power a

20. Advantages • Using expected value ofh resulted in excessively narrow h variation • Within dbh-class h variation resulted in wider h distribution • Improved goodness of fit • Example for pine • within dbh variation: • expected h = 22.5 to 26.0 m • ± 2 × sh h = 19.0 to 28.5 m

21. Within dbh-class h variation reasonable behaviour with respect to dbh and h more realistic description of the stand structure improve goodness of fit of the marginal h distribution slight improvement with wide dbh distributions (spruce) significant improvement with narrow dbh distributions and strongly bending h curve (pine) expexted h: 79% pass the K-S test Conclusions • including sh: • 98% pass the K-S test

22. will presumably benefit modelling increasingly complex stand structures Improved accuracy and flexibility in stand structure models