Things just don’t add up with SDI…

Martin Ritchie PSW Research Station

Overview

SDI: Concepts, LimitationsAdditivity: Stage (1968) Curtis/Zeide/Long Sterba & Monserud (1993)

Implications of maximum crown width functions…

Reineke (1933)

Reineke (1933)

Reineke's White Fir Plot

100

1000

10000

1 10 100

qmd

tpa

Problems with Reineke’s SDI

Definition of the “maximum” Applicable to even-aged standsNot 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)

Summation over all trees:

c = 1.605/2 = 0.8025

For individual-trees:

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)

Similar, in effect to traditional Tree-Area-Ratio approach for most diameters:

Tree-Area-Ratio (OLS) Approach using “well-stocked” unmanaged natural stands of Douglas-fir:

Zeide (1983)

Similar in form to Curtis (1971):

Proposed a different measure of stand diameter; a generalized mean with power = k.

Zeide (1983)

The modification of “mean stand diameter” results in an additive function for SDISDIz/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 skewnessp refers to slope of 1.6 for this case

Ratio of SDIz/SDIr

0.75

0.8

0.85

0.9

0.95

1

0 0.2 0.4 0.6 0.8 1

Coeff. of Variation

R

g=0

g=0

g=0.5

g=1

g=1.5

g=2

Long (1995)Additivity accounts for changes in stand

structure (empirically demonstrated):

0

50

100

150

200

250

0 2 4 6 8 10 12 1416 18 20 22 24 26 28 30

DBH (in)

Tre

es

per

Acr

e SDIr=927

SDIz=807

Problem:

100

1000

10 100stand diameter

2010 30

830740

tpa

Implies that the slope and the maximum remain constant with respect to changes in stand structure

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:

Diameter Distribution

0

5

10

15

20

25

30

4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

dbh

tpa

C=0.73

g=1.7

Uneven-aged Stand

4

4.5

5

5.5

6

2 2.2 2.4 2.6 2.8 3

slope= -1.490

Uneven-aged Stand

Ln(d)

Ln(tpa)

Diameter Distribution

0

5

10

15

20

25

4 5 6 7 8 9 10 11 12 13 14 15

dbh

tpa

Even-aged Stand

g=0.8

C=0.36

4

4.5

5

5.5

6

2 2.2 2.4 2.6 2.8 3

slope= -1.51

Even-aged Stand

Ln(d)

Ln(tpa)

4

4.5

5

5.5

6

2 2.2 2.4 2.6 2.8 3

Even-aged

Uneven-aged

Ln(d)

Ln(tpa)

4

4.5

5

5.5

6

2 2.2 2.4 2.6 2.8 3

Even-aged

Uneven-aged

sdi

Ln(d)

Ln(tpa).6*sdi max

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 applicationHowever, 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):93-96Reineke, 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. 16:403-414. 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