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Biomass
Equations
for
Intensively
Managed
Douglas-fir
Trees
David
W.
Hann,
Doug
Mainwaring,
and
Doug
Maguire
iomass
data needed
to
develop equations
are
difficult
and
expensive
to
col-
:
lect.
As a
consequence,
biomass data sets have been small
in
size
relative
to
the
size
of the
Douglas-fir population found
in the
Pacific
Northwest.
The
objective
of
this study
was to
create equation
forms
that would
be
more likely
to
accurately
and
precisely extrapolate
to
elements
of the
Douglas-fir population
not
found
in the
modeling
data set.
This
study
used
previous work
in
geometry
and
mechanics
to
develop interpretable nonlinear equation forms
for
predicting com-
ponents
of
Douglas-fir biomass using diameter
at
breast height
(D) in
cm,
total
tree
height
(H) in m, and
crown length (CL)
in
m
as
predictor variables.
Direct
predictors were developed
for the
following biomass components
in kg:
biomass
in
foliage
(FOL), living branches
(LIVEBR),
dead branches
(DEADER),
bark
(BRK),
sapwood
of the
stem (SAP), heartwood
of the
stem (HRT),
and
total
stem inside
bark
(TSB
). In
addition, indirect predictors, formed
from
direct predictors
of ap-
propriate components, were evaluated
for
total stem outside bark
(TSBob)
in kg and
total
tree living biomass above ground (TTLB)
in kg.
Data
A
total
of 200
trees were sampled from
23
stands that covered
a
range
in
ge-
ography,
allometry,
and
treatment history (thinning,
fertilization,
and
early weed
control).
The 23
sites included research installations (14)
and
operational timber-
land (9),
and
were distributed
from
42.80°
to
47.20°
N
latitude,
from
123.98°
to
121.67°
W
longitude,
and
from
140 to 790 m
above
sea
level. Measurements
of
FOL, LIVEBR,
BRK,
SAP,
and HRT
(TSB.b
being calculated
by
summing
SAP and
HRT) were made
on all 200
trees, while
DEADER
was
measured
on a
subsample
of 55
trees
for
which
51
were
in the
90th
percentile
of the
diameter distribution
and the
remaining
4
trees were
in the
10th percentile
of the
diameter distribution.
Details about sampling,
field and lab
work
are
available
in
Coons
et
al.
(2013).
Two
independent data sets were used
to
evaluate
the
capabilities
of
alternative
model
forms
to
extrapolate
to
different
combinations
of D, H, and CL and
differ-
ent
geographic locations than
found
in the
modeling data
set
(therefore,
they
are
not
considered validation
data
sets).
The first
data
comes from
a
single stand located
in
the
Siskiyou
Mountains
in
southwest Oregon (this will
be
called
the
SWO
data
set
in the
remainder
of
this report). Biomass
was
determined
for
FOL, LIVEBR,
BRK,
and
TSB.b
on 32
trees within
the
stand using
the
procedures described
in Nay
and
Bormann
(2014).
The
second data
set
comes from trees sampled
on
selected
old
growth stands
around
the
Shelton
Ranger District
of the
Olympic
National Forest
in
northwest
Washington (this will
be
called
the
NWW
data
set in the
remainder
of
this report).
In
this study, biomass
was
collected
by
Snell
and
Anholt (1981)
for
FOL, LIVEBR,
and
DEADER
on five
trees using essentially
the
same procedures
as
Brown (1978).
46
A
description
of the D,
H,
and CL
measurements found
in
all
four
data
sets
is
found
in
Table
1.
Table
1.
Summary
statistics
for
the
Douglas-fir
tree-level
biomass
modeling
data
set and the two
evaluation
data
sets.
Variable
Minimum
Mean
Maximum
Standard
Deviation
Modeling
Data
Set for
FOL,
LIVEBR,
BRK,
SAP,
and
HRT,
and
TSBib
(N =
200)
D
H
CL
D
H
CL
D
H
CL
D
H
CL
7.4
8.18
3.94
30.5
25.33
13.18
77.0
57.36
26.59
Modeling
Data
Set for
DEADER
(N = 55)
13.0
38.7 77.0
11.11
28.51
57.36
8.83
15.05
24.56
5WO
Evaluation
Data
(N
= 32)
23.1
45.9 80.3
20.1
29.1 36.0
8.5
15.8 27.6
NWW
Evaluation
Data
(N
=
5)
91.7
147.2
220.7
53.7
61.8 77.7
25.0
32.3 42.7
12.2
8.22
3.55
14.6
9.50
3.72
14.1
4.7
4.8
59.8
10.2
7.5
Methods
Equation
Forms
for
Predicting
Biomass
The
following equation forms were fashioned
from
equations developed
in
geometry
and
mechanics
in
order
to
predict biomass:
''
[1]
n)i.
=/?,
xj/nr/ior
xi;r/.
w)
[2]
/./r/-.«M=
A,
x[(LCW/lO)2x(PFOl
10)]'-
[3]
DKADHK
=
h{
x
<
/>
H
V';
*(!.<
'It'
1
0
r
x
(
Mlil.
•
1
0
)
[4]
TSH.f,
=/\
+/>,
x(/>
lOr
x(//.'
lO)xt.V<K
[5]
Mr-l'rS
[6]
IHU
»
I'l'S/t;h
X
( I -
RSAI')
[7]
HKK
-/>„
i
*,
xc/j/ior
x(//,
n))xi-*0f
[8]
l'SHllh
=
I'HRK
+
/'/.V%
[9]
ni.K=!'KRK
4
/'/\/%
4-
I'l-'Ot
+
ri.t\TJiR
Where,
LCW =
Largest crown width
of the
tree predicted
from
the
northwest
Oregon
Douglas-fir
equation
of
Hann
(1997)
in
m
PFOL
=
Predicted foliage weight
of the
tree
from
Equa-
tion
[1]
BL
= H - CL
MBL=
BL if BL
<
b2,
=
b2
if BL >
b2
CR =
CL/H
[10]
RSAP
=
1.0,
if(D
x H) I CR > 60,
otherwise
^.
RSAP-^xCR
+
V-Q-btxCRJe*1
in
PLIVEBR
=
Predicted LIVEBR
from
Equation
[2]
PTSB.b
=
Predicted
TSB.b
from
Equation
[4]
PBRK
=
Predicted
BRK
from Equation
[6]
Equations
[1],
[2],
[3],
[4],
[7],
and
[10]
are
consid-
ered
direct estimators
of the
response variables because
their parameters were estimated directly from
the
data.
Equations [5],
[6],
[8],
and [9] are
indirect estimators
of
the
response variables because they
use
only predictions
from
previously parameterized equations and, therefore,
do not
involve
the
direct estimation
of
parameters.
Estimation
and
Evaluation
Parameter estimates
for the
direct predictors
of
bio-
mass were estimated using weighted nonlinear regression
via
transformation.
Parameters
not
significantly
different
from
zero
at
(X
=
0.05 were eliminated.
The
quality
of the
resulting
fits was
analyzed
by
calcu-
lating
two
statistics
for
both
the
modeling data
set and for
the two
evaluation data sets:
(1) the
mean untransformed
residual
(predicted minus actual)
as a
percentage
of
mean
biomass (BIAS)
and (2) the
adjusted
coefficient
of
determi-
nation
of the
untransformed residuals
(UT_Rad.2).
While
the
parameter estimates
and
their standard errors were
determined using transformation
in
order
to
homogenize
variance
of the
residuals, normal application
of the
result-
ing
equations would produce untransformed predictions
so
this analysis
of the
quality
of fits was
done
on the un-
transformed residuals.
Results
and
Discussion
Parameter estimates
and
their standard errors
for
Equations [1], [2], [3], [4], [7],
and
[10]
are
found
in
Table
2. The
resulting
fit
statistics
for the
modeling data
set and
the
two
evaluation data sets
are
found
in
Table
3.
These
new
equations
for
directly predicting biomass
components were developed
from
prior
knowledge
of ge-
47
Table
2.
Parameter estimates
and
associated standard errors
of the
parameters
(in
parentheses)
for
predicting
Douglas-fir biomass
in Kg.
Parameters
(Standard
Errors)
Equation
b0
b,
b2
b,
[1]
[2]
[3]
[41
[71
HO]
NA
NA
NA
3.61578088
(0.58282)
0.318564479
(0.1583753)
NA
38.8305134
(0.9973194)
7.93338518
(0.3471377)
43.2444672
(6.92857)
14.1690124
(0.4942984)
2.60600933
(0.1496987)
0.362790893
(0.0352123)
NA
0.804701137
(0.01903674)
20.89
(5.169313)
-0.210021548
(0.0658482)
-0.385623122
(0.1097177)
-0.804811836
(0.02849096)
NA
NA
1.90718673
(0.3402741)
NA
NA
0.538867365
(0.01619606)
Table
3. Fit
statistics
BIAS
and
UW_y
for the
modeling
data
set and the two
evaluation data sets.
ometry
(e.g.,
FOL
Equation [1],
DEADER
Equation [3],
BRK
Equation [7],
and
TSB.b
Equation
[4])
or
basic
me-
chanics
(LIVEBR Equation
f2]).
Examination
of the fit
statistics
for the
modeling data
set
shows that
the
resulting
equations explained
69% to 89% of the
variation
in the
crown attributes
and 95% to 99% of the
variation
in the
stem attributes (Table
3).
Bias ranged
from
-2% to +5%
for
the
crown
attributes
and
from
-3% to +3% for the
stem
attributes.
The
lower
values
of UT R
.2
for the
crown
at-
—
ad[
tributes
is
probably
due to the
larger amount
of
unexplain-
able
measurement error
in
those response variables
due to
within
crown sampling.
The
objective
for
developing these
new
equations
was
to
explore model forms that might better characterize
the
biomass components
for
combinations
of
tree attributes
not
found
in the
modeling data set. Examination
of the fit
statistics
for the two
evaluation data sets
is
also shown
in
Table
3.
Equations
[1] and
[2]
were developed using
the
northwest Oregon version
of the LCW
equation.
The re-
sults
in
Table
3 for
these
two
equations applied
to the
SWO
evaluation data
set
shows that
the
southwest
Oregon
ver-
sion
of the LCW
equation provided substantially better
fit
statistics.
On the
other hand,
the
northwest
Oregon
LCW
equation
provided good
fit
statistics
for the NWW
evalua-
tion data.
These
results seem
to
imply, therefore, that,
for a
tree
of a
given DBH,
HT, and CL,
both
FOL and
LIVEBR
will vary
by
location
and
that
the LCW
equations
of
Hann
(1997)
are a
reasonable estimator
of
that
variation.
The
modeling data
set fit
statistics
for
DEADBR
(Equation
[3])
explained
the
least amount
of
variation
and
Equation
BIAS
UT-R42
Modeling
Data
Set
[11
[21
[3]
[4]
[5]
[6]
[7]
[81
[9]
-2.09%
-0.56%
+4.95%
+0.41%
+0.52%
+2.73%
-2.91%
+1.99%
+1.57%
0.8169
0.8928
0.6894
0.9920
0.9540
0.9636
0.9487
0.9920
0.9923
SWO
Evaluation Data
Set
[1]-NWOLCWEq.
[1]-SWOLCWEq.
[21-NWOLCWEq.
[2]
-SWO
LCW
Eq.
BRK
TSB,b
+14.9%
-9.1%
+68.5%
+15.5%
-7.0%
-3.7%
+0.6696
+0.8006
-0.8370
+0.6941
+0.9743
+0.9766
NWW
Evaluation Data
Set
l1]-NWOLCWEq.
[2]-NWOLCWEq.
[31-NWOLCWEq.
+1.2%
+9.5%
+578.0%
+0.7410
+0.7230
-54.3159
had the
largest amount
of
bias
for the
three crown attri-
butes
examined
in
this study (Table
3).
Equation
[3]
also
had
extremely poor
fit
statistics when applied
to the NWW
evaluation data set. Given
the
small
size
of the
modeling
data
set and the
concentration
of
that
sample
in
dominant
trees,
the use of
Equation
[3]
should
be
restricted
to the
population
found
in the
modeling data set.
The SWO
evaluation
data
set fit
statistics
for
theTSB.
ID
and BRK
equations (Equations
[4]
and
[7])
explained just
slightly less variation than
the
modeling data with slightly
increased bias (Table
3).
UT_Rad2
is
calculated using
the
mean square
of the
residuals
(MSE)
for the
evaluation data
sets
and
both
Dhrymes
et
al.
(1972)
and
Hocking
(1976)
have noted that this value must necessarily
be
larger
for an
evaluation
data
set
than
the MSE
found
for the
modeling
data set.
As a
result,
the UT R
,.2
of the
evaluation data
set
-
adj
must also
be
necessarily smaller than
the
UT_Rad
2
of the
modeling
data
set.
Therefore,
the
results
for
Equations
[1],
[4],
and [7]
indicate excellent performance when extrapo-
lated
to
southwest Oregon.
48
A
number
of
previous biomass studies have expressed
concern that
the
component biomass equations will
not
sum
up to an
unbiased predictor
of
total biomass (e.g.,
Chiyenda
and
Kozak 1984, Cunia
and
Briggs 1985,
Car-
valho
and
Parresol
2003).
As a
result, they have
proposed
procedures that constrain
the
parameters
of the
individual
equations
in
order
to
assure that they
sum up to an
unbi-
ased
estimator
of the
total biomass.
In
these procedures,
the
optimal properties
of the
individual equations
are
sacri-
ficed in
order
to
gain
the
property
of
additivity.
The
results
of
this study indicate
that
the
indirect estimators
of
TSBob
and
TTLB
are,
for all
practical purposes, unbiased (Table
3).
Therefore,
it is
concluded
that
carefully
developed
component equations
can
meet
the
property
of
additivity
without
the
need
to
compromise their individual optimal
statistical
qualities.
Literature
Cited
Brown, J.E. 1978. Weight
and
density
of
crowns
of
Rocky
Mountain conifers.
USDA,
Forest Service, Inter-
mountain Forest
and
Range Experiment Station,
Re-
search Paper
INT-197.
56p.
Carvalho, J.P.
and
B.R. Parresol.
2003.
Additivity
in
tree
biomass
components
of
Pyrenean
oak
(Quercus
pyre-
naica
Willd.).Forest
Ecology
and
Management 179:
269-276.
Chiyenda, S.S.
and A.
Kozak. 1984. Additivity
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compo-
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underly-
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K., D.
Maguire,
D.
Mainwaring,
A.
Bluhm,
R.
Har-
rison,
and E.
Turnblom. 2013.
Allometric
relationships
and
above-ground Douglas-fir biomass
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nutrient
pools
under varying stand density and nitrogen
fertil-
ization
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Pp.
28-32
in DA.
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and
D.B.
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291-324.
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D.W.
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predicting
the
largest
crown width
of
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western
Oregon.
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Corval-
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I4p.
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vari-
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S.M.
and
B.T. Bormann. 2014. Site-specific Doug-
las-fir
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the
Siskiyou
Moun-
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49
\*
•
^v*.
.i.*^,^*^
v
Annual Report 2014
•
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