Transcript Individual Tree Taper, Volume and Weight Loblolly and

Individual Tree Taper,
Volume and Weight
for Loblolly Pine
Bruce E. Borders
Western Mensurationists
Fortuna, CA
June 18-20, 2006
Current Models





Compatible total/merchantable tree
volume/weight/taper functions
Fitted separately by physiographic region (LCP, UCP,
Piedmont)
Large number of sample trees used in fit – however
the range of data is limited in stem size (approximately
1500 trees largest DBH = 14” class)
Problems using ratios low on the stem
Implied Taper functions not very realistic (taper
function derived from ratio volume equation)
Need More Data and Better
Models!


Large stumpage value drop for pulpwood in
many areas of the South (about 2000) have
resulted in more interest in solid wood
production
Hence, more users are finding the need to
merchandise stems into products that
require estimates for stem sections found in
the lower stem – current models do not
work well
Other Data Sources

CAPPS Destructively Sampled Trees



Age 6, 10, 12 years – data will be added to
PMRC individual tree database
Wood Quality Consortium Destructively
Sampled Trees – 272 trees distributed
across southern U.S.
U.S. Forest Service – FIA unit has a
relatively large database with volume,
weight and taper information available
Data Used to Fit Functions 2005
DBH
5
6
7
8
9
10
11
12
13
14
15
Total
Ht
20
30
40
50
60
70
80
90
Total
Trees
145
333
334
255
191
132
79
40
25
6
1
1541
Trees
3
185
389
412
333
160
58
1
1541
DBH
5
6
7
8
9
10
11
12
13
14
15
Total
20
2
30
68
91
26
1
3
185
40
54
154
128
44
7
2
389
Height (feet)
50
60
21
74
14
129
48
102
86
51
92
21
59
7
21
5
9
2
3
1
412
333
70
3
19
33
40
39
12
12
2
160
80
3
8
10
11
14
8
3
1
58
90
1
1
Total
145
333
334
255
191
132
79
40
25
6
1
1541
Data Used to Test Functions 2005
DBH
5
6
7
8
9
10
11
12
13
14
15
16
Total
Ht
20
30
40
50
60
70
80
90
Total
Trees
25
87
86
76
51
43
40
18
11
1
10
3
451
Trees
0
12
73
169
117
60
16
4
451
DBH
5
6
7
8
9
10
11
12
13
14
15
16
Total
20
30
3
9
40
16
31
16
8
1
1
Height (feet)
50
60
6
42
5
53
15
38
25
16
24
7
24
5
14
2
5
4
1
0
12
73
169
117
70
2
4
9
11
18
5
5
1
4
1
59
80
90
1
1
3
5
2
2
2
14
1
3
4
NOTE – all data will be combined for final model fits
Total
25
87
86
76
51
43
40
18
11
1
10
3
451
Existing Model Form
a4


D
a1
a2
m
V / W  a0 D H  a3  a4 2 H  4.5
D

 H h 
Dm  D

 H  4.5 
1
( a4  2 )
Where: V/W = cubic volume or weight to a top dob of Dm inches
H = total height (ft); D = dbh (inches)
Pienaar, L.V., T.R. Burgan and J.W. Rheney. 1987. Stem volume, taper and
weight equations for site-prepared loblolly pine plantations. PMRC Technical
Report 1987-1.
Existing Model Forms





Simple models – a lot of appeal for ease of
use
Models fitted separately by LCP, UCP,
Piedmont
Predict cubic volume inside/outside bark
Predict green weight inside/outside bark
Predict dry weight with and without bark
Existing Model Forms



These models were developed for use in
estimating volume or weight to a given top
diameter
However, the model form has limited
flexibility in reflecting stem form realistically
and it was fitted by Pienaar et al. (1987)
with data bases structured to have Dm
values of 6” and smaller
Today – users require capability to
merchandise stems into various products
that may be found anywhere within the stem
Existing Model Form

NOTE – the functions did not perform
any better when fitted to the new
database
New Functions –
Taper/Volume

Objective – provide two sets of functions
1.
2.
Simple and easy to implement (e.g. a ratio volume/weight
equation and associated taper function) – realize that
weaknesses will exist – fitted and evaluated model forms
suggested by Bailey (1994), Zhang et al. (2002) and Fang
et al. (2000) – only present results for Fang (2000) model
Sophisticated and very flexible system that should provide
very accurate estimates of stem volume and weight for
any stem segment – realize that implementation will
require thorough understanding of the functions to be
programmed (based on work of Clark, A. III, R.A. Souter and
B. Schlaegel. 1991)
New Functions - Weight
1.
2.
3.
Weights will be predicted using a per cubic
foot density measure (lbs of wood and bark
per cubic foot of wood)
Thus, to determine the green weight of
wood and bark in any specific stem section
we will calculate the cubic foot volume of
wood and multiply by lbs of wood and bark
per cubic foot of wood
These densities have been studied
extensively by Alex Clark and others.
Estimates currently available for different
regions by age classes.
Simple Models


Function flexibility and complexity increase
from Bailey (1994), Zhang et al. (2002) to
Fang et al. (2000).
Of course, stem shape is complex and
therefore it is not surprising the Fang et al.
performed best of these three alternatives

Fang, Z., B.E. Borders, and R.L. Bailey.
2000. Compatible volume-taper models for
loblolly and slash pine based on a system
with segmented-stem form factors. For. Sci.
46(1)
Simple Approach –
Alternative 3



In this work the stem profile was modeled
with 3 segments each with its own form
factor
The join points of the segments were
estimated as parameters
The model was derived to insure that the
taper function integrated to a total volume
that was consistent with a total stem volume
prediction equation that was fitted
simultaneously
Revised Fang et al. Model

I have revised the model as follows:
First join point is at 4.5 feet
 Taper function is constrained to go
through DBH
 No constraint to insure taper function
integrates to a given total volume
equation (as in the original paper)
 Form factors and upper join point vary
with tree dbh and height

Revised Fang et al. Model
d  c1  H

1 I1  I 2 
1
 
c1 
D
 H  4.5 
 
k  1
2 1
I1
2
I2
3
k  1
1
(1  p)   1I1  I 22I 2 

k 
k  576

1 p1  p  p2
I1  
0 otherwise
2  1  p2 
1 2
p
2
  3   2 k
  2  1 k
1  1  p1 
1
h
H
p1 
1 p2  p  1
I2  
0 otherwise
2 3
4.5
H
Revised Fang et al. Model
 1t0  I1  I 2 1   2 t1 
k 

2
1
V  c1 H  I 2 3   2 1t2

k


I1  I 2 I 2

  1  p  1 2

t0  1  p0 
k
1
p0 
stumpht
H
t1  1  p1 
k
1
t2  1  p2 
k
2
dod
1. 3
1. 2
1. 1
1. 0
0. 9
0. 8
0. 7
0. 6
0. 5
0. 4
0. 3
0. 2
0. 1
0. 0
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
hoh
oop
O
P
0. 8
0. 9
1. 0
1. 1
R
esi d_dob
4
3
2
1
0
-1
-2
-3
-4
-5
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
hoh
0. 7
0. 8
0. 9
1. 0
1. 1
R
esi d_vob
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
hoh
0. 7
0. 8
0. 9
1. 0
1. 1
dod
1. 3
1. 2
1. 1
1. 0
0. 9
0. 8
0. 7
0. 6
0. 5
0. 4
0. 3
0. 2
0. 1
0. 0
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
0. 7
hoh
oop
O
P
0. 8
0. 9
1. 0
1. 1
R
esi d_dob
2
1
0
-1
-2
-3
-4
-5
-6
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
hoh
0. 7
0. 8
0. 9
1. 0
1. 1
R
esi d_vob
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
0. 0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
hoh
0. 7
0. 8
0. 9
1. 0
1. 1
Revised Fang Model
Fang dod vs Rel Ht
1.2
D10_H60
0.8
D12_H60
0.6
D14_H60
0.4
D16_H60
0.2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Rel ht
Fang dod vs Rel Ht
1.2
1
dob/dbh
dob/dbh
1
D12_H50
0.8
D12_H60
0.6
D12_H70
0.4
D12_H80
0.2
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rel Ht
1
Revised Fang Model
Fang Model D=12 H=70
1.4
1.2
dib/dbh
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Rel Height
0.8
1
1.2
Not So Simple Approach (Because Tree
Shapes Are Not So Simple!!)
Clark, Souter and Schlaegel 1991
Souter and Clark 2001


Clark, A. III, R.A. Souter and B. Schlaegel. 1991.
Stem profile equations for southern tree species.
Research Paper SE-282. Asheville, NC. USDA
Forest Service, Southeastern For Exp Sta. 113 pp.
Souter, R.A. and A. Clark III. 2001. Taper and
volume prediction in southern tree species. USDA
For. Serv. Southern Research Station. FIA Work Unit
Administrative Report.
Souter & Clark Model



Three segment stem profile equation used
to define stem shape from ground line to
total height – each segment is fitted
independently of one another and are
constrained to be continuous at the join
points of 4.5’ and 17.3’
The first segment is divided into two subsegments to allow for more flexibility
The topmost segment is divided into three
sub-segments to allow for more flexibility
Souter & Clark Model
Function uses dbh, diameter at 17.3’
(Girard form class (GFC) height), and
total tree height
 Recall GFC = dib @ 17.3’/dbh
 Note – functions are provided to
predict dbhib from dbh and to predict
dob and dib at 17.3 as a function of
dbh and total height

D4.5
4.5'
h
hJ=4.5(1-J)
d
0'


d =  D 24.5









h<hJ IJ =1
D0=D4.5 (1+c)
.5
s

(1  I J )(1  c J (r - s)  4.45.5- h  ) +  

 
r

 
I J (1  c  4.45.5- h  )
 
Figure 3: Taper model for use at heights between ground line and 4.5'. Parameters, J, r, s, and c, are predicted as
functions of dbh (ob) and total tree height. Ground line diameter, D0, is not measured but predicted with the
flare parameter, c, and the provided diameter, D4.5.
D17.3
17.3'
h
d
4.5'
 
2
17.3- h 
d = D17.3

1  c17  17.3
 4.5 
D4.5=D17.3 (1+c17)
p

.5
Figure 2: Taper model for use at heights between 4.5' and 17.3'. The parameter, p, is predicted as a function of dbh
(ob) and total tree height. The flare parameter c17 is not predicted, but is calculated using the diameters at
4.5' and 17.3' that are provided, D4.5 and D17.3, respectively.
DH=0
H
h2=H-a2(H-17.3')
h1<h<h2 Ia2 =1
h1=H-a1(H-17.3')
h
h<h1 Ia1 =1
d
17.3'


 2
d =  D17.3



D17.3











I





( q1 - q2 )  H - h  q2



1
 H -17.3 


( q1- q2 )
( q 2 - q3 )  H - h  q3 


1
2
 H -17.3  
 H - h  q1
a1  H -17.3 
I a2 a
(1 - I a1 - I a2 ) a
+
+
a
.5







Figure 1: Taper model for use at heights between 17.3' and total tree height, H. Parameters, are predicted as functions
of observed dbh (ob) and total tree height. The diameter at 17.3', D17..3, is provided, and the diameter at total
height is assumed to be 0.
Souter & Clark Taper
Function



 H - h  q1


+
I a1  H - 17.3 





 2 
( q1 - q 2 )  H - h  q 2



 +
a
D
I
17.3
1
a

 if (17.3  h  H )
2
H
17.3







( q1 - q 2 )
( q 2 - q3 )  H - h  q 3 

 
 (1 - I a - I a ) a 1
a2
1
2
 H - 17.3  




p
 2
 17.3 - h 
d =  D17.3 1  c17  17.3  4.5 
if (4.5  h  17.3)





(r - s)  4.5 - h  s
 (1  I J )(1  c J

 )+ 
 4.5 
 2 

if (0  h  4.5)
D
4
.
5



r
4.5 - h


I J (1  c  4.5  )



























.5
eq .1
Souter & Clark Taper Function
Volume Equation
U
V = k L d (h) 2 dh

U H
V = k LJ0 d (h) 2 dh  H d (h) 2 dh  4.5 d (h) 2 dh  17.13 d (h) 2 dh  H 2 d (h) 2 dh  H
H
4.5
J
17.3
H
H
1
2
d (h) 2 dh

Souter & Clark Taper Function
Volume Equation
First calculate the join-point heights.
H J = 4.5(1  J )
H 1  H  a1 ( H  17.3)
H 2  H  a2 ( H  17.3)
Now determine segment limits of integration.
U J = min(max(U ,0), H J ) L J = min(U J , max(L,0))
U 1 = min(max(U , H J ),4.5) L1 = min(U 1 , max(L, H J ))
U 2 = max(min(U ,17.3),4.5) L2 = min(U 2 , max(L,4.5))
U 3 = max(min(U , H 1 ),17.3) L3 = min(U 3 , max(L,17.3))
U 4 = max(min(U , H 2 ), H 1 ) L4 = min(U 4 , max(L, H 1 ))
U 5 = max(min(U , H ), H 2 ) L5 = min(U 5 , max(L, H 2 ))
Souter & Clark Taper Function
Volume Equation
( r 1)
( r 1)
 2 

4.5c  4.5  LJ 
 4.5  U J   

 D4.5 (U J  LJ ) 



 

r

1
4
.
5
4
.
5








 


( s 1)
( s 1)




 2 

4.5c ( r  s )  4.5  L1 

 4.5  U1 


 

 D4.5 (U1  L1 )  s  1 J

4
.
5
4
.
5















( p 1)
( p 1) 


12
.
8
17
.
3

L
17
.
3

U




2
2
2
2
2
 D (U  L )  ( D  D )







4.5
17.3
 17.3 2 2
p  1  12.8 
 12.8 
 
V=k 

 2 H  17.3  H  L3  ( q1 1)  H  U 3  ( q1 1) 


D







 17.3

q1  1  H  17.3 
 H  17.3 





( q 2 1)
( q 2 1)

 H U4 
 D 2 H  17.3 a ( q1  q 2 )  H  L4 




17.3
1


q2  1
 H  17.3 
 H  17.3 



(
q

1
)
(
q

1
)
3
3
 2 H  17.3 ( q  q ) ( q  q )  H  L 


 H  U5 
5
a1 1 2 a2 2 3 

 D17.3




q3  1
 H  17.3 


 H  17.3 

eq .2
Souter & Clark Taper Function
Height Prediction for Given Top Diameter


1
2

q

 d  3
 H  H  17.3 a1q2  q1  a 2q3  q2   2  



 D17.3  


1

2
q

 2
 H  H  17.3 a1q2  q1   d  
 2 


 D17.3  


1

2
q1
 H  H  17.3 d 
 2 

 D17.3 
h= 
1

2
2
17.3  12.8 d  D17.3  p
 2  2 

 D 4.5 D17.3 

1

s
2
4.5  4.5   d  1 1 
2
(r s )

  D4.5  cJ


1

1r
 2
4.5  4.5   d2  1 
  D4.5  c 

0




q1  q2  q2
2
2

if D17.3 a1
a2  d  0



q1
 q1  q2  q2
2
2
2

if D17.3 a1  d  D17.3 a1
a2



q1
2
2
2

if D17.3  d  D17.3 a1




2
if D 24.5  d 2  D17
.3 



r
2

if D24.5 (1  c J )  d  D 24.5 



2
r
2
2
if D4.5 (1  c )  d  D 4.5 (1  c J ) 


if d 2  D 24.5 (1  c ) 







eq.3
Souter & Clark Taper Function
Auxiliary Variable Prediction
Auxiliary variables used to estimate taper include:
1)
2)
3)
4)
5)
Dbh (ib) from dbh (ob), (dibdbh, dobdbh)
Diameter at 17.3' (ib) from diameter at 17.3' (ob), (dib173, dob173)
Diameter at 17.3' (ob) from diameter at 17.3' (ib), (dob173, dib173)
Diameter at 17.3' (ib) from dbh (ob), and total height, (dib173, dobdbh, tht)
Diameter at 17.3' (ob) from dbh (ob), and total height, (dob173, dobdbh, tht)
Souter & Clark Taper Function
Auxiliary Variable Prediction
With model forms, respectively,
1) dibdbh=adibdbh+bdibdbh*dobdbh+adibdsz*dsize+bdibdsz*dsize*dobdbh;
dsize is an indicator for sawtimber sizes; =1 if dbh class>=9" for softwoods, >=11" for hardwoods.
Parameters are adibdbh, bdibdbh, adibdsz, and bdibdsz.
2) dib173=adibf+bdibf*dob173+adibfsz*dsize+bdibfsz*dsize*dob173;
Parameters are adibf, bdibf, adibfsz, and bdibfsz.
3) dob173=adobf+bdobf*dib173+adobfsz*dsize+bdobfsz*dsize*dib173;
Parameters are adobf, bdobf, adobfsz, and bdobfsz.
4) dib173=dbh/(1+exp(-(edib17th+(cdib17th+ndib17th*indicsd)*dbh
+ldib17th*log(tht)**2+mdib17th*indicsh +ddib17th*log(tht)+idib17th*dbh*log(tht))));
size indicator used are indicsh=(tht<30); indicsd=(dbh<=8);
Parameters are edib17th, cdib17th, ndib17th, ldib17th, mdib17th, ddib17th, and idib17th.
5) dob173=dbh/(1+exp(-(edob17th+(cdob17th+ndob17th*indicsd)*dbh
+ldob17th*log(tht)**2+mdob17th*indicsh+ddob17th*log(tht)+idob17th*dbh*log(tht))));
Parameters are edob17th, cdob17th, ndob17th, ldob17th, mdob17th, ddob17th, and idob17th.
Souter & Clark Taper Function
Auxiliary Variable Prediction
• To implement – predict dbhib from dbh
(eq. 1); predict dib17.3 from dbh (eq.
4); predict dob17.3 from dib17.3 (eq.
3) – do not use eq. 2 or 5.
 Any parameters shown in equations
above that do not appear in the
parameter estimate lists below should
be set to 0
Weight Density

Clark, A. III, R.F. Daniels and B.E. Borders.
2005. Effect of rotation age and
physiographic region on weight per cubic
foot of planted loblolly pine. Southern
Silvicultural Confernce. Memphis, TN.
March 2005.
Weight Density
All individual tree weight data from the
PMRC, WQC and U.S. Forest Service
was used for this work
 Loblolly plantations were separated
into two regions – Atlantic and Gulf
Coastal Plains combined and
Piedmont, Upper Coastal Plain and
Hilly Coastal Plain (Inland) combined

Weight Density
Weight Density

Three age classes
10 to 18 years
 19 to 27 years
 > 27 years

Weight Density

Green weight of wood and bark per
cubic foot of wood – can use in
conjunction with inside bark cubic
volume functions shown above to
obtain estimated weight of wood and
bark
Weight Density
Coastal Plains – 68.12 lbs wood and
bark/cubic foot of wood (66.91 to
69.33)
 Inland – 66.61 lbs wood and
bark/cubic foot of wood (65.89 to
67.32)

Weight Density

Lbs wood and bark/cubic foot of wood
Region
Coastal
Coastal
Coastal
Inland
Age Class
10 – 18
19 – 27
>27
10 – 18
Mean
70.87
67.69
65.28
69.20
LCL
69.21
67.16
63.67
66.30
UCL
72.52
68.22
66.89
70.10
Inland
19-27
67.03
66.50
67.56
Inland
>27
64.60
63.44
65.77
Weight Density

Reasons why this density decreases as
trees age:


As dbh increases (as trees age) the
proportion of stem weight in bark decreases
(thus denominator (cubic feet of wood) tends
to be larger for older trees)
Wood moisture content decreases with
increasing tree age (averaged 124% for 14
yr old trees, 114% for 24 yr old trees, 104%
for 34 yr old trees)
Weight Density


Further work is underway to look at defining
density of green weight of wood/cubic foot
of wood and dry weight of wood per cubic
foot of wood
Also – developing functions to predict these
density measures for different tree ages
along the stem and how best to use these
functions in conjunction with the
taper/volume functions
Summary
Bottomline – several improved
functions available for taper/cubic
volume/weight determination for
loblolly pine plantations – user’s
choice (if Clark et al. model is not
used the best choice is the revised
Fang model)
 Same work will be done for slash pine

THE END
Remember – it is
important to get out
of the truck every
now and then!
Revised Fang OB Fit
The MODEL Procedure
Nonlinear SUR Summary of Residual Errors
Equation
dm
vol
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
4.5
4.5
14777
14777
2489.8
5851.7
0.1685
0.3960
0.4105
0.6293
0.9757
0.9886
0.9757
0.9886
Nonlinear SUR Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
pp1
pp2
pp3
bb1
bb2
mm1
mm2
mm3
bet3
-13.3103
4.304361
-0.34946
0.001325
-0.00004
0.002128
5.103E-6
-0.00001
0.001899
0.4499
0.1364
0.0139
0.000017
1.955E-6
0.000013
2.853E-7
1.531E-6
9.892E-6
-29.58
31.56
-25.08
78.70
-19.26
169.33
17.88
-9.11
191.95
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
p2 = 1/(1+exp(-(pp1 + pp2*log(tht) + pp3*dbh)));
bet1 = bb1 + bb2*dbh;
bet2 = mm1 + mm2*tht +mm3*dbh;
Revised Fang IB Fit
The MODEL Procedure
Nonlinear SUR Summary of Residual Errors
Equation
dm
vol
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
4.5
4.5
14777
14777
2172.6
5062.0
0.1470
0.3426
0.3834
0.5853
0.9737
0.9868
0.9737
0.9868
Nonlinear SUR Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
pp1
pp2
pp3
bb1
bb2
mm1
mm2
mm3
bet3
-8.22247
2.602632
-0.22236
0.001468
-0.00006
0.002367
6.763E-6
-0.00003
0.001869
0.2223
0.0647
0.00720
0.000018
2.293E-6
0.000017
3.767E-7
2.146E-6
7.901E-6
-36.99
40.23
-30.90
79.73
-24.25
137.41
17.95
-12.60
236.53
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
p2 = 1/(1+exp(-(pp1 + pp2*log(tht) + pp3*dbh)));
bet1 = bb1 + bb2*dbh;
bet2 = mm1 + mm2*tht +mm3*dbh;
Clark & Souter Taper Function
Taper Parameters
Each parameter is predicted with unique sets of coefficients, ( ,  , ), with some of the estimated coefficients being 0.
Parameters and model forms associated with Figure 1:
q1=(aq1+bq1*(tht)+cq1*dbh);
a1=1/(1+exp(-(aa1+ba1*log(tht)+ca1*dbh)));
q2=(aq2.+bq2*(tht)+cq2*dbh);
a2=(1/(1+exp(-(aa2+ba2*log(tht)+ca2*dbh))))*a1;
q3=(aq3+bq3*(tht)+cq3*dbh);
Parameters and model form associated with Figure 2:
p=exp(ap+bp*(tht)+cp*dbh);
Parameters and model forms associated with Figure 3:
J=1/(1+exp(-(aJ+bJ*log(tht)+cJ*dbh)));
s.=(as+bs*(tht)+cs*dbh);
r=(ar+br*(tht)+cr*dbh);
c=(ac+bc*log(tht)+cc*dbh);
While the above systems of taper parameters are described for outside bark diameter measurements, completely
analogous parameters are used for inside bark diameter predictions, where dbh in the equations would refer to an inside
bark dbh. Parameters would be subscripted with "o" or "i" to indicate the appropriate system.
NOTE – any parameters that do not appear in the estimate lists below
should be set to 0.
Clark & Souter Taper Function
Taper Parameters - ib
The MODEL Procedure
Nonlinear OLS Summary of Residual Errors – Base Segment
Equation
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
5
10161
61.4121
0.00604
0.0777
0.5880
0.5878
DODI
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
ASI
AJI
ARI
ACI
BCI
1.24775
1.218833
4.086773
1.080878
-0.04477
0.0857
0.0501
0.0538
0.0682
0.0173
14.56
24.35
75.95
15.84
-2.58
<.0001
<.0001
<.0001
<.0001
0.0098
The MODEL Procedure
Nonlinear OLS Summary of Residual Errors – 4.5 to 17.3’
Equation
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
3
8861
8.5015
0.000959
0.0310
0.8446
0.8446
DODI
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
API
BPI
CPI
-0.41234
0.004642
0.065145
0.0228
0.000744
0.00578
-18.06
6.24
11.28
<.0001
<.0001
<.0001
Clark & Souter Taper Function
Taper Parameters - ib
The MODEL Procedure – 17.3’ to Tip
Nonlinear OLS Summary of Residual Errors
Equation
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
12
18978
29.8629
0.00157
0.0397
0.9619
0.9619
DODI
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx
Std Err
AQ1I
CQ1I
AQ2I
BQ2I
CQ2I
AQ3I
BQ3I
CQ3I
AA1I
BA1I
AA2I
BA2I
1.111066
-0.01107
2.480819
-0.04164
0.162101
1.884622
-0.01296
0.080945
12.55434
-2.85796
-12.0778
3.192642
0.0225
0.00250
0.0374
0.00116
0.00642
0.0534
0.00110
0.00509
0.3893
0.0947
0.9261
0.2285
t Value
Approx
Pr > |t|
49.46
-4.43
66.40
-35.96
25.24
35.29
-11.80
15.91
32.25
-30.19
-13.04
13.97
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Souter & Clark Taper Function
Predict dbhib from dbh
The GLM Procedure
Dependent Variable: DIBDBH
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
1
8736.638139
8736.638139
95266.6
<.0001
Error
2644
242.473898
0.091707
Corrected Total
2645
8979.112037
R-Square
Coeff Var
Root MSE
DIBDBH Mean
0.972996
4.554590
0.302832
6.648942
Source
DF
Type I SS
Mean Square
F Value
Pr > F
DOBDBH
1
8736.638139
8736.638139
95266.6
<.0001
Source
DF
Type III SS
Mean Square
F Value
Pr > F
DOBDBH
1
8736.638139
8736.638139
95266.6
<.0001
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Intercept
DOBDBH
-.4344503820
0.9180106242
0.02369246
0.00297425
-18.34
308.65
<.0001
<.0001
Souter & Clark Taper Function
Predict dib173 from dbh
The MODEL Procedure
Nonlinear OLS Summary of Residual Errors
Equation
DIB173
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
7
2684
360.3
0.1342
0.3664
0.9612
0.9611
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
cdib17th
ddib17th
edib17th
idib17th
ldib17th
mdib17th
ndib17th
-0.32344
8.990871
-18.01
0.075332
-1.04591
-0.10512
-0.00662
0.0641
0.8436
1.4740
0.0157
0.1195
0.0421
0.00205
-5.04
10.66
-12.22
4.79
-8.75
-2.50
-3.23
<.0001
<.0001
<.0001
<.0001
<.0001
0.0127
0.0013
Souter & Clark Taper Function
Predict dob173 from dib173
The MODEL Procedure
Nonlinear OLS Summary of Residual Errors
Equation
DOB173
DF
Model
DF
Error
SSE
MSE
Root MSE
R-Square
Adj
R-Sq
4
2687
105.5
0.0393
0.1981
0.9900
0.9900
Nonlinear OLS Parameter Estimates
Parameter
Estimate
Approx
Std Err
t Value
Approx
Pr > |t|
adobf
bdobf
adobfsz
bdobfsz
0.223475
1.048978
0.183364
-0.01595
0.0200
0.00434
0.0472
0.00698
11.15
241.70
3.88
-2.28
<.0001
<.0001
0.0001
0.0225