Transcript Additivity and SDI

Things just don’t add up with
SDI…
Martin Ritchie
PSW Research Station
Overview
SDI: Concepts, Limitations
Additivity:



Stage (1968)
Curtis/Zeide/Long
Sterba & Monserud (1993)
Implications of maximum crown width
functions…
Reineke (1933)
Reineke (1933)
Reineke's White Fir Plot
tpa
10000
1000
100
1
10
qmd
100
Problems with Reineke’s SDI
Definition of the “maximum”
Applicable to even-aged stands
Not Additive:

No way to determine contribution of
individual trees or cohorts to total SDI
Implied Comparability
SDI is meaningful as long as the
maximum for a given comparison is a
constant.
SDI is meaningful as long as the slope
is fixed (you can’t compare across
slopes).
Stage (1968)
Suppose the following relationship holds for
individual-tree sdi:
Stage (1968)
Stage (1968)
This reduces to one variable: c=1.605/2,
which acts as a weight on diameter-squared
Stage (1968)
For individual-trees:
Summation over all trees:
c = 1.605/2 = 0.8025
Stage (1968)
There are an infinite number of
solutions for “c” between 0 and 1 which
will solve the equation.
c=.8025 is not necessarily optimal…
e.g., c=1, then a=0 and b=SDI/BA.
Curtis (1971)
Tree-Area-Ratio (OLS) Approach using “well-stocked”
unmanaged natural stands of Douglas-fir:
Similar, in effect to traditional Tree-Area-Ratio
approach for most diameters:
Zeide (1983)
Proposed a different measure of stand diameter; a
generalized mean with power = k.
Similar in form to Curtis (1971):
Zeide (1983)
The modification of “mean stand
diameter” results in an additive function
for SDI
SDIz/SDIr=f(c.v.)
Zeide (1983)
Taylor Series Expansion about the arithmetic
mean for the Generalized Mean:
1st
3rd
4th
C = coefficient of variation
g = coefficient of skewness
p refers to slope of 1.6 for this case
Ratio of SDIz/SDIr
1
g=0
g=0
g=0.5
0.95
g=1
g=1.5
g=2
R
0.9
0.85
0.8
0.75
0
0.2
0.4
0.6
Coeff. of Variation
0.8
1
Long (1995)
Additivity accounts for changes in stand
structure (empirically demonstrated):
Trees per Acre
250
200
SDIr=927
SDIz=807
150
100
50
0
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
DBH (in)
Problem:
Implies that the
slope and the
maximum remain
constant with
respect to changes
in stand structure
1000
830
740
tpa
100
10
10
20
stand diameter
30
Sterba and Monserud (1993)
Slope is a function of stand structure
(skewness):
-Slope decreases as the skewness of the
stand increases.
-Change in slope is substantial
Sterba & Monserud (1993)
Additivity is effective within stand
structure…
Difficult to make comparisons between
stands of different structures…
So what?
Does the relationship really change with
maximum or the slope?
Open Grown Trees
Using MCW=f(dbh),
And, some known distribution, with g
fixed calculate an implied constant
density line:
Uneven-aged Stand
Diameter Distribution
30
25
C=0.73
tpa
20
g=1.7
15
10
5
0
4
6
8
10
12
14
16
18
20
dbh
22
24
26
28
30
32
34
36
Uneven-aged Stand
6
5.5
slope= -1.490
5
Ln(tpa)
4.5
4
2
2.2
2.4
2.6
Ln(d)
2.8
3
Even-aged Stand
Diameter Distribution
25
20
C=0.36
tpa
15
g=0.8
10
5
0
4
5
6
7
8
9
10
dbh
11
12
13
14
15
Even-aged Stand
6
5.5
slope= -1.51
5
Ln(tpa)
4.5
4
2
2.2
2.4
2.6
Ln(d)
2.8
3
6
Even-aged
Uneven-aged
5.5
5
Ln(tpa)
4.5
4
2
2.2
2.4
2.6
Ln(d)
2.8
3
6
Even-aged
Uneven-aged
5.5
sdi
5
Ln(tpa)
.6*sdi max
4.5
4
2
2.2
2.4
2.6
Ln(d)
2.8
3
Conclusions
Stage’s Solution to additivity is not unique,
may or may not be optimal.
Long’s conclusion with uneven aged stands
may be naïve, because maximum may
change with changes in structure.
Slope may change as well (Sterba &
Monserud), causing problems with application
However, MCW functions imply consistency
across diameter distributions with a stable
slope near Reineke’s 1.6 and a stable
maximum for ponderosa pine.
References
Curtis, R.O. 1971. A tree area power function and related stand density measures
for Douglas-fir. For. Sci. 17:146-159.
Long, J.N. 1995. Using stand density index to regulate stocking in uneven-aged
stands. P. 111-122 In Uneven-aged management: Opportunities, constraints and
methodologies. O’Hara, K.L. (ed.) Univ. Montana School of For./ Montana For. And
Conserv. Exp. Sta. Misc. Publ. 56.
Long, J.N. and T.W. Daniel. 1990. Assessment of growing stock in uneven-aged
stands. West. J. Appl. For. 5(3):93-96
Reineke, L.H. 1933. Perfecting a stand-density index for even-aged forests. J.
Agric. Res. 46:627-638.
Shaw J.D. 2000. Application of stand density index to irregularly structured stands.
West. J. Appl. For. 15(1):40-42.
Sterba, H. and R.A. Monserud. 1993. The maximum density concept applied to
uneven-aged mixed species stands. For. Sci. 39:432-452.
Sterba, H. 1987. Estimating potential density from thinning experiments and
inventory data. For. Sci. 33:1022-1034.
Stage, A.R. 1968. A tree-by-tree measure of site utilization for grand fir related to
stand density index. USDA For. Serv. Res. Note INT-77. 7 p.
Zeide, B. 1983. The mean diameter for stand density index. Can. J. For. Res.
13:1023-1024.
Some Other Interesting SDI-Related Stuff
Chisman, H.H. and F.X. Shumacher. 1940. On the tree-area ratio and certain of
its applications. J. For. 38:311-317.
Curtis, R.O. 1970. Stand density measures: an interpretation. For. Sci. 16:403414.
Lexen, B. 1939. Space requirements of ponderosa pine by tree diameter. USDA,
Forest Service, Southwestern Forest and Range Experiment Station Res. Note 63. 4
p.
Mulloy, G.A. 1949. Calculation of stand density index for mixed and two aged
stands. Canada Dominion Forest Serv. Silv. Leaflet 27. 2p.
Oliver, W.W. 1995. Is self-thinning in ponderosa pine ruled by Dendroctonus bark
beetles? In: Eskew, L.G. comp. Forest health through silviculture. Proceedings of the
1995 National Silviculture Workshop; 1995, May 8-11; Mescalero New Mexico.
General Technical Report RM-GTR-267. Fort Collins CO: USDA, Forest Service, Rocky
Mountain Forest and Range Experiment Station. 213-218.
Schnur, G.L. 1934. Reviews. J. For. 32(3):355-356.
Spurr, S.H. 1952. Forest Inventory. Ronald Press, New York. Pages 277-288