Improving the accuracy of predicted diameter and

height distributions

Jouni SiipilehtoFinnish Forest Research Institute, Vantaa

E-mail: jouni.siipilehto@metla.fi

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 dgM, hgM

– age (T) and basal area (G)

• Number of stems (N) is additional character, which is not required

Objectives

• 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

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

Diameter distribution

• The three-parameter Johnson’s SB distribution – bounded system includes the minimum and the

maximum endpoints – the minimum of the SB distribution () was fixed

at 0 – fitted using the ML method– to describe the basal-area diameter distribution

(DDG )

– transformed to stem frequency distribution (DDN)

Distribution function

• Johnson’s SB 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

25,0exp2

1dd zzdf

d

dz

dd

dddd

ln

ddzd

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 (ModelG)

– or those with the addition of a stem number (N) observation (ModelGN)

”Percentile method”

• When predicting the SB distribution,

parameter was solved according to known and and median dgM using Formula

• Thus, known median was set for predicted distribution.

gMgM dd lnˆˆlnˆ

”Shape index”

• Single stand variables: dgM, G, N or T did not

correlate closely with the shape parameter of the SB distribution

• In ModelGN, stand characteristics were linked together for ”shape index”

– in which

Ng

G

M

21004 gMM dg

The behaviour of the shape index ψ

0 20 40

0.05

0.1Shape=0.94

d, cm

P

0 20 40

0.05

0.1Shape=0.89

d, cm

P

0 20 40

0.05

0.1Shape=0.76

d, cm

P

0 20 40

0.05

0.1Shape=0.74

d, cm

P

0 10 20 30

0.05

0.1Shape=0.51

d, cm

P

0 20 40

0.05

0.1Shape=0.38

d, cm

P

Stem frequency (solid line) and basal area distributions (dotted line)

Correlation between parameter and shape index for spruce and pine

• Correlation r = 0.57 and 0.68 for pine and spruce, respectively

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5

delta

shap

e in

dex

Spruce

Pine

Results: Prediction models

• ModelG – dgM and T explained , and stem form (dgM/hgM) was

the additional variable explaining – r2 for and

• 0.22 and 0.05 for pine• 0.40 and 0.28 for spruce

• ModelGN – Shape index alone or with dgM explained and – r2 for and

• 0.28 and 0.38 for pine• 0.37 and 0.50 for spruce

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

ModelModelGG ModelModelGNGN

PinePine BiasBias ssbb BiasBias ssbb

V 3.0 5.1 2.4 4.9

N -4.8 12.6 -4.4 6.1

SpruceSpruce

V 1.7 6.0 2.2 5.4

N 8.7 25.0 -6.0 12.3

The predicted DDGs (above) and the derived DDNs for spruce and pine, when

1.0, 0.77 and 0.63Spruce

0

0.02

0.04

0.06

0.08

0.1

0 10 20 30 40

P(g

)

0

10

20

30

40

50

60

0 10 20 30 40

d, cm

n

Pine

0

0.05

0.1

0.15

0 10 20 30 40

1.00

0.77

0.63

0

5

10

15

20

25

30

0 10 20 30 40

d, cm

Advantages

• ModelGN is capable of describing great variation in N within fixed dgM and G

• Example– dgM=20 cm, G=20 m2ha-1

if = 1.00 then N = 705 and 790 ha-1

if = 0.63then N = 1020 and 1100 ha-1

for pine and spruce, respectively

Unbiased N = 640 and 1020 ha-1

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

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

3.1

10

d

dh

d

h

d101

3.1

Error structure handling

• The residual variation (sz) of from linearized model

• transformation to concern real within-dbh-class height variation (sh)

• using Taylor’s series expansion

d

h

sszh

1

3.1ˆ

Error structure behaviour

Pine

0

5

10

15

20

25

30

0 10 20 30 40

d, cm

h, m

•funtion of diameter and height

•dependent on height curve power

Spruce

0

5

10

15

20

25

30

0 10 20 30 40

d, cm

h, m

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 × sh h = 19.0 to 28.5 m

0

5

10

15

20

25

30

35

0 10 20 30 40

d, cm

h, m

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

Improved accuracy and flexibility in stand structure

models

will presumably benefit modelling increasingly complex stand structures