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

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

3. Objectives The objective of this study: –to examine whether the accuracy of predicted basal-area diameter distributions (DD G ) could be improved by using stem number (N) together with basal area (G) –in terms of degree of determination (r 2 ) –in terms of stem volume (V) and total stem number (N), when –G is unbiased

4. Study material Study material consisted of: –91 stands of Scots pine (Pinus sylvestris L.) –60 stands of Norway spruce (Picea abies Karst.) both with birch (Petula pendula 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

5. Diameter distribution The three-parameter Johnson’s S B distribution –bounded system includes the minimum and the maximum endpoints –the minimum of the S B distribution (  ) was fixed at 0 –fitted using the ML method – to describe the basal-area diameter distribution (DD G ) –transformed to stem frequency distribution (DD N )

6. Distribution function Johnson’s S B distribution is based on transformation to standard normality in which - z is standard normally distributed variate  and  are shape parameters  and are the location and range parameters -d is diameter observed in a stand plot

7. Predicting the distribution Species-specific models for predicting the SB distribution parameters  and Linear regression analysis The models were based on either –predictors that are consistent with current FMP (Model G ) –or those with the addition of a stem number (N) observation (Model GN )

8. ”Percentile method” When predicting the S B distribution, parameter  was solved according to known  and and median d gM using Formula Thus, known median was set for predicted distribution.

9. ”Shape index” Single stand variables: d gM, G, N or T did not correlate closely with the shape parameter  of the S B distribution In Model GN, stand characteristics were linked together for ”shape index”  –in which

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

11. Correlation between parameter  and shape index  for spruce and pine Correlation r = 0.57 and 0.68 for pine and spruce, respectively

12. Results: Prediction models Model G –d gM and T explained, and stem form (d gM /h gM ) was the additional variable explaining  –r 2 for and  0.22 and 0.05 for pine 0.40 and 0.28 for spruce Model GN –Shape index  alone or with d gM explained and  –r 2 for and  0.28 and 0.38 for pine 0.37 and 0.50 for spruce

13. b The relative bias and the error deviation (s b ) of the volume and stem number in the test data Model G Model GN PineBias sbsbsbsbBias sbsbsbsb V3.05.12.44.9 N-4.812.6-4.46.1 Spruce V1.76.02.25.4 N8.725.0-6.012.3

14. The predicted DD G s (above) and the derived DD N s for spruce and pine, when  1.0, 0.77 and 0.63

15. Advantages Model GN is capable of describing great variation in N within fixed d gM and G Example –d gM =20 cm, G=20 m 2 ha -1 if  = 1.00 then N = 705 and 790 ha -1 if  = 0.63 then N = 1020 and 1100 ha -1 for pine and spruce, respectively Unbiased N = 640 and 1020 ha -1

16. Height distribution Height distribution is not modelled for FMP purposes It is produced with a combination of dbh distribution and height curve models –only expected value of height is used for each dbh class –height distribution has become of great interest lately from stand diversity point 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

17. Height model including error structure Näslund’s height curve Linearized form for fitting –power  =2 and 3 for pine and spruce respectively –  0 and  1 estimated parameters Residual error  : –homogenous variance –normally distributed

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

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

20. Advantages Using expected value of h 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 × s h h = 19.0 to 28.5 m

21. Conclusions 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 including sh: –98% pass the K-S test

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