improving the accuracy of predicted diameter and height distributions jouni siipilehto finnish...
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![Page 1: Improving the accuracy of predicted diameter and height distributions Jouni Siipilehto Finnish Forest Research Institute, Vantaa E-mail: jouni.siipilehto@metla.fi](https://reader035.vdocuments.us/reader035/viewer/2022070400/56649f115503460f94c24aa1/html5/thumbnails/1.jpg)
Improving the accuracy of predicted diameter and
height distributions
Jouni SiipilehtoFinnish Forest Research Institute, Vantaa
E-mail: [email protected]
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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
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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
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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
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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)
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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
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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)
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”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
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”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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Improved accuracy and flexibility in stand structure
models
will presumably benefit modelling increasingly complex stand structures