Modeling crown rise in even-aged stands of Sitka spruce or loblolly pine

Forest Ecology and Management, 69 (1994) 189-197

189

Elsevier Science B.V.

Modeling crown rise in even-aged stands of Sitka spruce or loblolly pine Harry T. Valentine~, Anthony R. Ludlow b, George M. FurnivaF aUS Dep. Agric. Forest Service, Louis C. Wyman Forest Sciences Laboratory, PO Box 640, Durham, N i l 03824-0640, USA bForestry Commission, Forest Research Station, Alice Holt Lodge, Farnham, GU10 4LH, UK cYale University, School of Forestry and Environmental Studies, 360 Prospect Street, New Haven, CT 06511, USA (Accepted 5 August 1992 )

Abstract

A 'crown-rise' model that estimates average height to the base of a crown in an even-aged, monospecific stand is derived and fitted to loblolly pine and Sitka spruce data. Estimated standard errors are less than 1 m. The driving variables are average tree height and either tree count per unit area or average inter-tree distance. Two potential uses of the crown-rise model are: ( 1 ) a component of an empirical or mechanistic forest model, and (2) an alternative to stocking charts for stand density management.

Introduction In recent years, forest scientists have begun to develop mechanistic, process-based models of tree or stand growth. When deriving a mechanistic model, a scientist must decide how to parametrize the key physiological processes of a tree as functions of the rather limited number of tree variables that can be measured with available instrumentation. One obvious key physiological rate is photosynthesis, which would be parametrized most naturally in terms of leaf area or dry matter, or some surrogate of leaf quantity such as cross-sectional area of the bole at the crown base, sapwood area, crown diameter, or crown length. According to pipe-model theory (Shinozaki et al., 1964), the leaf dry matter on a healthy tree is highly correlated with cross-sectional area of the bole Correspondence to: H.T. Valentine, US Dep. Agric. Forest Service, Louis C. Wyman Forest Sciences Laboratory, PO Box 640, Durham, N H 03824-0640, USA.

© 1994 Elsevier Science B.V. All rights reserved 0378-1127/94/$07.00 S S D I 0 3 7 8 - 1 1 27 ( 9 2 ) 0 3 14 1-2

190

H.T. Valentine et al. / Forest Ecology and Management 69 (1994) 189-197

at the base of the crown (A). Based on studies of bole taper in many species (e.g. Gray, 1956 ), we infer that A ~-BR, where B is cross-sectional area of the bole at breast height, and R is the ratio of the length of the live crown to the length of the tree above breast height. To support this hypothesis, we analyzed data from 172 spruce-fir (Picea spp., Abies balsamea (L.) Mill. ) trees from Maine and obtained a slope of c = 1.01 for the regression model A=cBR (r2> 98% ). Thus, it appears that we can use BR as a surrogate for leaf quantity for some modeling applications (e.g. pipe models of spruce or fir growth ), but in so doing, we will need to estimate R. The use of either R or the conventional crown ratio (crown length/tree height) as a state variable in forest models is neither new nor limited to mechanistic models. For example, crown ratio is used in the 'prognosis model' (Stage, 1973 ), the STEMS model (Belcher et al., 1982 ), and in weighted constructions of APA polygons (Nance et al., 1988). It is not surprising that crown ratio has been used in empirical models; it is perhaps more surprising that crown ratio, or some absolute measure of crown size, has not been used in most of them. As Furnival (1987) noted, crown ratio clearly separates understocked, unthinned stands with low basal area from heavily stocked stands that are thinned back to the same basal area. It also can be a useful index of taper and quality, i.e. knot size and number. Modeling crown ratio or some other crown variable can be accomplished in several ways. Mitchell ( 1975 ) simulated entire crowns, but for many modeling problems this level of detail is neither necessary nor desirable. In this article, we present a detailed description and analysis of a simple 'crown-rise' model that furnishes the average height to the base of a crown in an evenaged, monospecific stand. The crown-rise model can be used in connection with either empirical or mechanistic, distance-independent, tree-level, forest models. It also easily adapts for use in distance-dependent models. Combined with a model that furnishes tree height, the crown-rise model provides time streams of both crown length and crown ratio. The crown-rise model was first formulated by G. Furnival in 1972 and used in an unpublished forest model. An equivalent crown-rise model was independently formulated and used by Ludlow et al. (1990) in a Sitka spruce (Picea sitchensis (Bong.) Carr.) model. Beekhuis ( 1965 ) developed a model of canopy depth in radiata pine (Pinus radiata D. Don), which is similar to the crown-rise model in some aspects. The crown-rise model

We consider an even-aged, monospecific stand with a population density

of Nt trees per hectare in Year t. The average land area of a tree is Nt -~ and, assuming regular spacing, the average distance between trees (/)t) in meters is:

H. 12 Valentine et al. /Forest Ecology and Management 69 (1994) 189-197

191

(1)

15, = ( l o O00/Nt) '/2

If land area is measured in acres, and/9, in feet, then 43 560 replaces I0 000 in eqn. ( 1 ). We denote the average height of a tree by/-7, and the average height to the base of a live crown by C,. Both heights have the same dimension as Dt. Average crown length equals/-it- C,. We begin by formulating the crown-rise model for the closed stand. We assume, for the moment, that ( 1 ) the trees have conical crowns (like those of Sitka spruce), and (2) the average basal diameter of a crown equals the intertree distance,/St. Under these assumptions, the crown length of a tree with average dimensions is: (2)

I-It--<=~°St

and the height to the base of the crown of the tree is:

C,=II,-fl.D,

(3)

where/~ is the ratio of average crown length to average crown diameter. In reality, of course, the crowns of most species are not cones and average crown diameter may not equal/St. However, eqns. (2) and ( 3 ) also hold under the more general assumption that the average crown length is proportional to average inter-tree distance, regardless of crown shape or average diameter. We proceed under this more general assumption. For diagrammatic purposes, we express the parameter fl of eqns. (2) and (3) in terms of an angle (0), where O= arctan

[Dt/(Ht -

Ct) ].

(4)

Thus, fl= 1/tan O, and our crown-rise model for the closed stand, eqn. (3), becomes: Ct = / t t - (D,/tan 0)

(5)

The relation between 0, average crown length, and average inter-tree distance is depicted in Fig. 1 (a). The behavior of the crown-rise model for the closed stand is quite simple. If inter-tree distance remains constant from one year to the next, average crown length also remains constant because the crown-rise increment equals the average height increment (i.e. C t - Ct_ 1= I ~ t - - I - t t - 1 ). If inter-tree distance increases from self-thinning, the average crown length of the residual trees also increases because the crown-rise increment is less than the average height increment by the amount (/~t-/~t-- 1 ) / t a n 0. However, if the stand is opened by a silvicultural thinning, eqn. (5) no longer applies. As shown in Fig. 1 (b) a heavy silvicultural thinning creates the situation where average crown length is shorter than it could be, given the new intertree distance (i.e. aqt - C, < LSt/tan 0). Thus Ct> H t - / ) t / t a n 0. We assume ( 1 )

192

H. Z Valentineet al. / Forest Ecology and Managernent 69 (1994) 189-197

.......

j ///i

S

/]

l/

I " ......

Closed stand: C't =/t,

i

- (Dt/tanO)

\-----5 J - - - /ti - (/)l/tanO) /)i

~'-

Thinned,

open stand: C'l = C'i-1

I¢/

__

I

I

bi

---f--II

fli-(bi/tanO)

New, open stand: C'i = 0 General model: C'l = max[0, C't-1,/~i - (hi~ tan 0)]

Fig. 1. Diagram of the crown-rise model for (a) closed stands, (b) thinned, open stands, and (c) new, open stands. ~t is average tree height, Ct is average height to the base of a crown,/)t is average inter-tree (bole to bole) distance, and 0 is a constant angle. The appropriate model for Ct is given for each type of stand.

that the average height to the base of a crown remains stationary (i.e. Ct = Ct_ ~) as long as the stand remains open. This implies that average crown length increases at the rate of average height growth. We also assume (2) that closure occurs and crown rise resumes when average height increases to the point where average crown length equals/)t/tan 0. These assumptions also

H. T, Valentine et al. / Forest Ecology and Management 6 9 (1994) 189-197

193

apply to a newly established, weed-free plantation, where Co=0 (see Fig.

l(c)). Combining our assumptions for open stands with those for the closed stand, we obtain: C~=max [ 0 , C t _ , , / t , - (Dt/tan 0)].

(6)

This general crown-rise model applies whether the stand is closed, thinned, or new.

Analysis The only parameter of the crown-rise model, 0 (or fl), pertains to closed stands. We obtained data to estimate 0 from loblolly pine (Pinus taeda L. ) stands in Texas, and from a Sitka spruce thinning trial in Wales. The data collectors did not indicate which data were obtained from closed stands in either place. Therefore, our first task was to determine which stands were closed from an examination of the data. We then used closed-stand data to fit: C, = / 4 , - (/gt/tan 0)+%

(7)

with non-linear regression techniques (Wilkinson, 1988 ). We treated re-measurements of stands like independent measurements. As an alternative to the non-linear techniques, we could have used linear regression techniques to fit:

(8) which would furnish 0= arctan/~- 1.

Loblolly pine The loblolly pine data were supplied by J. David Lenhart from the East Texas Pine Plantation Research Project. The data were gathered from 175 plots, each 0.093 ha (0.23 acres). Height and height to the base of the crown were measured on each tree in each plot. Most plots were measured three times on a 2 or 3 year interval. Many of the plots were from young stands that had not closed. In some of these stands, Co was recorded as 0. In others, however, (70 was recorded as greater than or equal to 1 ft (0.3 m), reflecting either different data-gathering protocols or premature crown rise caused by weed competition. In most of the stands that had closed, self-thinning was nil to light. Figure 2 shows arctan Dr~ (I-it-Or) plotted against ~ for all loblolly pine stands and measurement times. Since none of the stands had been thinned, we assumed that all points to the right of the dashed line in Fig. 2 (Ct>_ 6 ft ( 1.83 m) ) obtained from closed stands that had suppressed their herbaceous weed competition. These (n = 211 ) data were used to fit eqn. ( 8 ), which fur-

194

H.T. Valentine et al. / Forest Ecology and Management 69 (1994) 189-197

f),ll(H, - C',)

Arctan 9O

(deg)

o o o

?%o

60

O0

0

I

~000 o ../d~J

OI

00 O0

0 I oo

~o~o~

o

, o~

~

~L

o

00

30

o

~

o ~ ! ~ ^ o ~ ~ o ~ d ~ _

o

o

o °

~3& ~

.~

o

oO

~0

o o

0

o

o O~

....

o oO0 ^

o O

.

0

~o ~ " o oo V o ° ~o8°O I

0

o

0 o

oo o

o

o

o

I I I I

I

I

I

I

0

I

I

I

3

I

I

I

6

I

I

112

9

C', (m) Fig. 2. Arctan D t / ( H , - C , ) versus Ct for 175 loblolly pine stands measured one to three times each. Data points which fall to the right of the dashed line were assumed to derive from closed stands. These data were used to estimate 0, the expected value of arctan Dt/(Ht-Ct) in the closed stands. The resultant estimate, 0, is depicted by the solid line. &

15-

oO//

o od°Xo o

10

15

10 o

0

0

~, (m) 5

o ooOO -

I

I

1

I

I

I

I

I

5

I

I

10 C, ( m )

I

I

I

I

I

15

I

0

0

I

1

I

I

I

I

I

I

5

I

10

I

I

I

I

I

15

C, (m)

Fig. 3. Predicted average height to the base of a crown (Ct=Ht-Dt/tan O) versus true average height to the base of a crown (C) for presumably closed stands of (a) loblolly pine in which Ct> 1.83 m (6 ft) and (b) Sitka spruce, where the base of a crown is defined by the lowest live branch.

195

H.T. Valentine et al. / Forest Ecology and Management 69 (1994) 189-197

nished 0 = 2 6 . 0 ° (95% confidence interval: 25.5 ° < 0< 26.5 °; standard error of an estimate of Ct: 0.87 m (2.84 It) (see Fig. 3(a) ). Sitka spruce The Sitka spruce data were gathered from a thinning trial initiated by the British Forestry Commission in Dyfi, Wales. Twenty-four plots were subjectcd to thinnings of various intensities and frequencies; three of these plots were not thinned. Three to ten trees from each plot were measured at irregular intervals between the ages of 17 and 38 years. These measurements included height, height to the lowest live branch, height to the lowest live whorl, and crown width. Estimates of plot averages calculated from these measurements have associated sampling errors. a

b 16

o

o

12

/

12 o o

o

o

o

o

Average crown

o

8 O°o°

o oo

length

o

Oo

(m)

o

0

0

I

0

I

I

I

4

b,(m)

I

I

I

I

8

0

I

I

I

I

I

4

b,(m)

Fig. 4. Average crown length versus average inter-tree distance,/St, in re-measured Sitka spruce plots. For the left graph (a), crown length was measured from the lowest live branch to the tree tip. The fitted line intersects the ordinate axis at angle ~= 20.6 °. For the right graph (b), crown length was measured from the lowest live whorl to the tree tip. The fitted line intersects the ordinate axis at angle ~= 25.35 °.

196

n.T. Valentine et aL / Forest Ecology and Management 69 (1994) 189-197

We classified a given plot at a given time as open if average crown width was less than average inter-tree distance, 15,. We pooled the (n = 67 ) data from the closed plots and found three apparent outliers in a graph of average crown length versus inter-tree distance (Fig. 4). To avoid undue influence of the outliers on our estimates of 0, we used the robust regression strategy of minimizing the sum of l etl (e.g. Weisberg, 1985 ), rather than the sum ofe~, to fit eqn. (7). Defining C, as the average height to the lowest live branch obtained 0= 20.6 ° (95% confidence interval: 20.1 o < 0< 21.1 °; average absolute error of an estimate of Ct: 0.83 m (2.74 ft) (see Fig. 3 ( b ) ) . Defining C~ as the average height to the lowest live whorl obtained 0=25.35 ° (95% confidence interval: 24.8 ° < 0< 26.9 °; average absolute error of an estimate of Ct: 0.79 m (2.59 ft) ). Discussion We have demonstrated that our simple crown-rise model fits data from closed loblolly pine and Sitka spruce stands reasonably well, but the fitting procedure does not constitute a rigorous test of the model. Some rigorous testing will be carried out in connection with the continuing development of the mechanistic Sitka spruce model described by Ludlow et al. (1990), but further testing for loblolly pine and other southern yellow pines is needed. There is also the question of whether the model will perform adequately for other species, particularly broad-leaved species. We encourage anyone with data to address this question to do so. However, some elaboration of the crown-rise model may be needed. Size-related changes in crown morphology or shade tolerance may cause 0 to vary in closed stands of some species. One obvious elaboration would be to formulate 0 as a function of height, substitute the formulation for 0 into eqn. (7), and fit the whole model with nonlinear techniques. There is also the possibility that 0 may vary with site. When applying the crown-rise model to a species with conical crowns, it is convenient to express the parameter of the model in terms of the angle ( a ) formed by the intersection of the edge of the crown with the main stem at the apex of a tree of average dimensions, i.e. a = arctan/5t/2 (Hi - Ct ). Thus, the conical crown-rise model is: Ct=max[0, G - l , / - / t - ( / 5 t / 2 tan a ) ] , where 2 tan a = t a n 0= 1/fl. This 'conical formulation', which is used in the Sitka spruce model of Ludlow et al. (1990), affords direct and easy measurement of the parameter, i.e. the crown angle a. This feature may prove useful for applications in high latitudes, where conical crowns are most prevalent, and crown angle seems to vary with the sun angle. Regardless of whether our particular model proves generally applicable, we expect a great deal of advancement in crown models in the next few years. This should benefit silviculturists as well as mechanistic modelers. It is no secret that trees with relatively short crowns produce relatively little volume

1t. T. Valentine et al. /Forest Ecology and Management 69 (1994) 189-197

197

increment. Thinnings are used to increase the crown sizes of the residual trees so they can fix more carbon and produce more increment. Some of the most successful thinning regimes tend to keep a nearly closed canopy with as many leaves as possible on as few trees as possible. Designing such a regime is not easy. Many silviculturists use a stocking chart or density index to determine whether a stand needs thinning. However, stocking charts are not based on crown dynamics. A stocking chart indicates when to thin an 'average stand', but the stand of interest may not be average. A crown-rise model, on the other hand, might be used to determine whether the crowns in a particular stand have attained their maximum length, given the population density, or whether, for example, tree height must increase by 3 m before maximum crown length is attained and increment begins to decline. Thus, one potential payoff of crown models for silviculture is customized stand management.

References Beekhuis, J., 1965. Crown depth of radiata pine in relation to stand density and height. NZ. J. For., 10: 43-61. Belcher, D.M., Holdaway, M.R. and Brand, G.J., 1982. STEMS - - the stand and tree evaluation and modeling system. Gen. Tech. Rep. NC-79. US Dep. Agric. For. Serv., North Cent. For. Exp. Stn., St. Paul, MN, 18 pp. Furnival, G.M., 1987. Growth and yield prediction: some criticisms and suggestions. In: H.N. Chappell and D.A. Maguire (Editors), Predicting Forest Growth and Yield: Current Issues, Future Prospects. Contribution No. 58. Inst. For. Resources, University of Washington, Seattle, pp. 23-30. Gray, H.R., 1956. The form and taper of forestqree stems. Imp. For. Inst. No. 32. Oxford University, 79 pp. Ludlow, A.R., Randle, T.J. and Grace, J.C., 1990. Developing a process-based model for Sitka spruce. In: R.K. Dixon, R.S. Meldahl, G.A. Ruark and W.G. Warren (Editors), Process Modeling of Forest Growth Responses to Environmental Stress. Timber Press, Portland, OR, pp. 249-262. Mitchell, K.J., 1975. Dynamics and simulated yield of Douglas-fir. For. Sci. Monogr. 17, 39 pp. Nance, W.L., Grissom, J.E. and Smith, W.R., 1988. A new competition index based on weighted and constrained area potentially available. In: A.R. Ek, S.R. Shirley and T.E. Burk (Editors), Forest Growth Modelling and Prediction. Vol. 1. Gen. Tech. Rep. NC-20. US Dep. Agric. For. Serv., North Cent. For. Exp. Stn., St. Paul, MN, pp. 134-142. Shinozaki, K., Yoda, K., Hozumi, K. and Kira, T., 1964. A quantitative analysis of plant form: the pipe model theory. II. Further evidence of the theory and its application in forest ecology. Jpn. J. Ecol., 14(4): 133-139. Stage, A.R., 1973. Prognosis model for stand development. US Dep. Agric. For. Serv. Res. Pap. INT-164. Intermt. For. Range Exp. Stn., Ogden, UT, 32 pp. Weisberg, S., 1985. Applied linear regression. John Wiley, New York, 324 pp. Wilkinson, L. 1988. SYSTAT: The System for Statistics. SYSTAT, Inc., Evanston, IL, 822 pp.