Transcript Ei dian otsikkoa - Luonnonvarakeskus

Improving the accuracy of
predicted diameter and
height distributions
Jouni Siipilehto
Finnish Forest Research Institute, Vantaa
E-mail: [email protected]
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 (x) 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
f d   zd

1
2
exp  0,5 zd
2

• is based on
transformation to
standard normality
 d  xd 

zd  g d  d d ln
 ld  x d  d 
• in which
zd 
-
dl
d  x l  x  d 
z is standard normally
distributed variate
 g and d are shape
parameters
 x and l are the location
and range parameters
- d is diameter observed in
a stand plot
Predicting the distribution
• Species-specific models for predicting
the SB distribution parameters d 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)
”Percentile method”
• When predicting the SB distribution,
parameter g was solved according to known
d and l and median dgM using Formula


g  dˆ ln lˆ  d gM  dˆ lnd gM 
• Thus, known median was set for predicted
distribution.
”Shape index”
• Single stand variables:
dgM, G, N or T did not
correlate closely with
the shape parameter d
of the SB distribution
• In ModelGN, stand
characteristics were
linked together for
”shape index” y
G
y
gM N
– in which
gM 

4
d
100 
2
gM
The behaviour of the shape index ψ
Stem frequency (solid line) and basal area distributions (dotted line)
Shap e=0 .94
20
0.05
40
0
d, cm
0
40
20
d, cm
20
30
Shap e=0 .3 8
0.1
0.05
0
10
d, cm
P
P
P
20
d, cm
40
Shap e=0 .7 4
0.1
0.05
0
0.05
d, cm
Shap e=0 .8 9
0.1
20
Shap e=0 .5 1
0.1
P
0.05
0
Shap e=0 .7 6
0.1
P
P
0.1
40
0.05
0
20
d, cm
40
Correlation between parameter d and
shape index y for spruce and pine
1.0
0.8
shape index
• Correlation r
= 0.57 and
0.68 for pine
and spruce,
respectively
0.9
0.7
0.6
Spruce
0.5
Pine
0.4
0.3
0
1
2
3
delta
4
5
Results: Prediction models
• ModelG
– dgM and T explained l, and stem form (dgM/hgM)
was the additional variable explaining d
– r2 for l and d
• 0.22 and 0.05 for pine
• 0.40 and 0.28 for spruce
• ModelGN
– Shape index y alone or with dgM explained l
and d
– r2 for l and d
• 0.28 and 0.38 for pine
• 0.37 and 0.50 for spruce
The relative bias and the error deviation (sb) of
the volume and stem number in the test data
ModelG
Pine
Bias
ModelGN
sb
Bias
sb
V
3.0
5.1
2.4
4.9
N
-4.8
12.6
-4.4
6.1
Spruce
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
y1.0, 0.77 and 0.63
Spruce
0.1
Pine
0.15
1.00
P(g)
0.08
0.77
0.1
0.06
0.63
0.04
0.05
0.02
0
0
0
10
20
30
10
20
10
20
d, cm
30
40
30
60
25
50
n
0
40
40
20
30
15
20
10
10
5
0
0
0
10
20
d, cm
30
40
0
30
40
Advantages
• 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
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 canopydwelling 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 a =2 and 3 for pine
and spruce respectively
– b0 and b1 estimated
parameters
• Residual error e:
– homogenous variance
– normally distributed
a
d
h
 1.3
a
b 0  b1d 
d
h  1.3
a 1
 b 0  b 1d  e
Error structure handling
• The residual variation (sez) of
e from linearized model
• transformation to concern real
within-dbh-class height
variation (seh)
• using Taylor’s series
se h
expansion


a 1
 ˆ

a
a  h  1.3 


 se z
d
Error structure behaviour
•funtion of diameter
and height
•dependent on height
curve power a
Pine
30
30
25
25
20
20
h, m
h, m
Spruce
15
15
10
10
5
5
0
0
0
10
20
d, cm
30
40
0
10
20
d, cm
30
40
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
35
30
25
h, m
•
20
15
10
5
0
0
10
20
d, cm
30
40
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:
•including sh:
– 79% pass the K-S test
–98% pass the K-S test
Improved accuracy and
flexibility in stand structure
models
will presumably benefit modelling
increasingly complex stand structures