modeling forest canopy heights: the effects of canopy shape

ELSEVIER

Modeling Forest of Canopy Shape

Canopy Heights: The Effects

R o s s N e l s o n °

Tree-dimensional models that represent the top-@canopy fi~rest height structure were developed to simulate airborne laser p@l ing responses along forested transects. The simulator which produced these 3-D nmdels constructed individual tree' crowns according to a tree's total height, height to first branch, crown diameter, and crown shape (cone, parabola, ellipse, sphere, or a random (tssortment of these shapes'), and then inserted these crowns into a fixed-area plot using ,napped stand (x,y) coordinates. This two-dimensional array of forest canopy heights" was randomly transected to simulate measurements' made by an airborne ranging laser. These simulated laser nwasurements were regressed with ground reference nw(t~ures to dewelop predictive linear relation- ships. The assumed crown shape had a significant impact on 1) simulated laser measurements' of height and 2) estimates of basal area, woody volume, and above-ground drp bionuts's derived via s'imulation. As" canopy shape progress'ed from a conic .titan to a more .spheric structure, average canopy height, canopy profile area, and canopy wdume increased, canopy height variation decreased, and coefficients of variability were stable or decreased. In Costa Rican tropical forests', simulated laser measurements of average height, canopy profile area, and canopy volume increased 8-10% when a parabolic rather than a conic shape was assunwd. An elliptic canopy was 16-18% taller, on average, than a conic canopy, and a .spheric canopy was 23-25% taller. The effect of these height increases and height variability changes can pro- fimndly affect basal area, >olume, and biomass estimates, but the degree to which these estimates are affected is study-area-dependent. Since canopy shape may sign{fi- cantly c{~ect such estimates, canopy shapes should be

°Biospheric Sciences Branch, NASA/Goddard Space Flight Cen- ter, Greenbelt

Address correspondence to Ross Nelson, Biospherie Seienees Br., NASA/GSFC, Code 923, Greenbelt, MD 20771.

Receired 29 Januar~j 1996; revised 5 October 1996.

REMOTE SENS. ENVIRON. 60:327-334 (1997) ©Elsevier Science Inc., 1997 655 Avenue of the Americas, New York, NY 10010

noted when field data are collected fi~r purposes of height simulation. I f canopy shapes are not noted and are un- known, an assumption of an elliptical shape is suggested in order to mitigate potentially large errors which may be incurred using a generic assumption of a cone or sphere. © Elsevier Science Inc., 1997

I N T R O D U C T I O N

The use of airborne lasers to estimate remotely Jbrest re- sources entails a procedure where airborne laser measurements of forest height and ground measurements (e.g., woody biomass) are related empirically (Nelson et el., 1984; 1988a,b; Aldred and Bonner, 1985; Maelean and Krabill, 1986). Predictive regressions based on a portion of the airborne laser data set are applied to all of the airborne data acquired over the study area to develop regional estimates. The compilation of the data needed to compute the prediction equations has, to this point, been contingent upon the aeeurate location of portions of the airborne laser transeet on the ground. Colocation has been and is difficult. The coregistration of the laser and ground lines is most difficult in remote, traekless re- gions where an airborne laser might be applied to best effect--areas such as the Amazon, the Congo, and the boreal forests.

Nelson et el. (1997) report a procedure that may be used to cireumvent this requirement to register the airborne laser transects on the ground. The procedure in- volves a computer simulation of the height characteristics of the ibrest canopy based on ground plot data where stem locations and canopy characteristics (e.g., total tree height, canopy diameter, height to first branch) are recorded. The digital canopy heights of the 3-D forest model are sampled to simulate airborne laser transects. If tree canopy shapes are not recorded in the field (mid they were not in this study), then shape nmst be assumed to conduct the simulation. That assumption affects airborne laser estimates of woody biomass since

003,t-4257/97/$17.09 PII S0034-4257(96)00214-3

3 2 8 Nelso,~

canopy shape affects the height characteristics of the simulated stand (Solodukhin et al., 1977; 1985), which, in turn, form the basis of predietive regression equations relating laser measurements to ground measurements of interest.

The enmneration of the effects of canopy shape are important because shape affects simulated estimates of average height and height variability. Since basal area, woody volume, and biomass have been modeled as linear functions of these variables (Maclean and Krabill, 1986; Nelson et al., 1988a,b; Nelson, 1994), assumptions of canopy shape have a direct effect on the accuracy of laser inventory estimates. The objective of this investiga- tion is to quantitatively characterize the effects of the canopy shape assumption on such laser measurements as average canopy height, height variability, and on the basal area, volume, and biomass estimates derived from these laser measurements.

PROCEDURE

A five-step procedure was used to assess the effects of different canopy shape assumptions on estimates of basal area, bole volume, and above-ground dry biomass.

Step 1. Acquisition of Airborne Laser Data Airborne laser profiling measurements were acquired in October 1984 over two sites in Costa Rica (CR). See Nelson et al. (1977, beginning of the second section) fbr details concerning the location and attributes of the airborne laser data sets.

Step 2. Acquisition of Ground Reference Data Mapped stand data were acquired on fixed area plots in the proximity of but not coincident with the airborne laser lines. Descriptions of the three ground data sets-- Tileran, LaSelva-5 m, and LaSelva-20 m--may be found in Nelson et al. (1997, Ground Data Collection subsec- tion in the second section).

Step 3. Development of a Three-Dimensional Forest Canopy Model Canopy heights were simulated for 50 m segments of all fixed-area ground plots [see Nelson et al. (1977) this issue, for justification of the 50 m segment length]. A two- dimensional array of tree heights was created on a 0.25 m grid such that each element of the array represented the height of the forest canopy above a 0.25 m×0.25 m section of the ground plot. The 10 m×500 m Tileran ground plot, for instance, was divided into 10~50 m segments, with the tree heights on each segment represented in a 40×200 element array. On each segment, 30 random lines were drawn along the long axis of the segment to simulate linear samples of laser canopy height measurements. Averaged simulated laser measurements

Table 1. Equations Used To Calculate Tree Canopy Heights at Diflbrent (x,y) Locations in an Individual Tree (Jrown Based on Total Tree Height, Height to First Branch, and Crown Diameter

Conic

Parabolic

Elliptic

Spheric

: ~ = h , - ( h ' - h t ' ) ~ - r , ) '-

L E ~ j

See also Nelson and Yaya (1990) for these equations. All variable units of length, including x and y, are identical. Variables: (x,y)=grid location of the point under consideration; distance along- and across- transect froin the ground transect origin, z~=height of the crown at a particular x,y location; if solution to equation is negative, z, is set to zero. h,=total tree height, h/,=height to first branch, r,=crown radius, one- half of the crown diameter.

such as average height of all "pulses," average height of canopy, variability measures, and canopy closure were calculated based on those 30 lines. Associated ground measures of basal area, volume, or biomass were also noted tbr each segment. These simulated laser measurements should have been comparable, ideally, to what an airborne laser would have measured if the flightline had traversed the same stand(s) of trees. Realistically, airborne laser measurements differed from simulated measurements due to simplifying assumptions employed to conduct the simulation, including the following:

1. A tree's canopy was circular when viewed from above.

2. A tree's canopy was centered on the bole of the tree, as viewed from above.

3. A particular tree's canopy, in profile, was assumed to be one of four geometric shapes--a cone, pa- raboloid, ellipsoid, or spheroid. Random noise (distributed N(0, 0.12 me), Nelson, 1994) was added for surfhce texture.

Table 1 reports the equations used to develop the canopy heights for the individual trees in the ground plot data. These trees, once constructed, were inserted into the transect array." centered about the x,y location of the trees' boles. The conic, parabolic, and elliptic canopies extended symmetrically from the top of the tree to the height of the first branch. The spheric canopies extended symmetrically from the top of the tree until the canopy diameter limit was reached; the height to first branch was not considered. Nelson and Yaya (1990) illustrated the canopy construction procedure and reported the ef- teets of canopy shape on simulated laser measurements. Simulated canopy heights for a 100 m section of the second of the LaSelva-20 m ground plots are illustrated in Figure 1.

Telnplates tbr the canopy shapes considered in this study may be found in textbooks that cover analytic ge-

Modeling Forest Canopy Heights: Effects of Canopy Shape 329

ometry, including most calculus textbooks. Illustrations of different geometric shapes as they apply to tree canopies were also reported by Mawson et al. (1976), who ac- tually characterized a tree canopy as a function of two geometric shapes, a profile shape (cross-sectional shape from the side)--circle, triangle, neiloid, parabola, ellipse--and a plan shape (cross-sectional shape from the top)----circle, ellipse, triangle. Their tree canopy constructs, then, were more sophisticated than ones used in this study, in part because they were interested in characterizing leafy vob ume rather than characterizing that volume between the top of the canopy and the ground.

Step 4. Development of Regression Models Regressions were developed that related ground-measured basal area, volmne, or biomass to the simulated laser measurements. Regression models of the following form (Nelson, 1994) were calculated using the simulation results from step 3. All linear models were forced through the origin.

b~=af .,+af,,,

b~=af~+a2c~,

G =alh,: + a,2c,, + a3c,,,

t)t=(l lhc + a 2ca + a3Cc,

where

b~,bt=basal area, seen, ~ and basal area, total (m2/ha), v~,vt=bole volume, seen, and bole volume, total (ran/ha)

for the Tileran study site; or above-ground dry biomass, seen, and above-ground dry biomass, total (mtons/ha) for the LaSelva study sites,

h,, = average of all of the simulated laser height measurements along a particular segment, including ground hits (m),

/~,,= average of only those simulated height measurements along a particular segment which intercept tree canopies (m) [ground hits (h~=0) are excluded],

c,,,c, = corresponding coefficients of variation.

Fifteen sets (5 canopy shapes×3 study areas) of these tbur regressions were calculated. The three study areas included the Tileran I0-m-wide data set, the LaSelva 5-m-wide data set, and the LaSelva 20-m-wide data set. The five canopy shapes tested include cone, parabola, ellipse, sphere, and a random assignment of these four shapes.

A "seen" tree is one whose crown fbrms a portion of the top of the forest canopy in the three dimensional computer simulation. "Seen" refers to a particular tree eanopy's potential to be sensed or directly illuminated by a vertieally pointed airborne laser. The basal area, vohlme, and biomass of the forest is divided into its seen and unseen compo- nents. Seen+unseen=toted. "Seen" is roughly equivalent to oversto~y.

Step 5. Airborne Laser Estimates of Basal Area, Volume, Biomass Study area estimates of seen and total basal area, volume, or biomass were generated using the predictive equations from step 4 in conjunction with the airborne laser data (step 1). Intereomparison of basal area, volume, or biomass estimates among canopy shapes for a given study area were done to quantify the et}~ets of shape assumptions. No attempt was made to identify a "correct" or most accurate canopy shape since other fac- tors [discussed in Nelson et al. (1997)] have a much larger impaet on estimation accuracy. Rather, the objective is to define quantitatively the degree to which a particular assumed canopy shape alters predicted seen and total basal area, volume, or biomass relative to other shapes' estimates.

RESULTS

Table 2 reports the effects of canopy shape on average height and height variability of tile simulated forest canopies. These data represented averaged values over all simulated plots; individual plot results were weighted by length in the case of the LaSelva-5 m and LaSelva-20 m data sets. As can be seen in Table 2, the average simulated height values increased as the shape varied from a conic section to spheric. The parabolic canopy shape was approximately 8.6-10.0% taller than a canopy comprised of cones, the elliptical canopy was 16.5-18.0% taller than the conic canopy, and the spheric canopy was 23.7- 24.6% taller than cones. A random selection of these four canopy shapes yielded a result midway between the conic and spheric results. Note that canopy profile area (p, in m2), that is, that vertical area described between the laser canopy and ground traces, and canopy volume (~ ...... in m:~), that is, that volume between the top of the forest canopy and the ground, react exactly as h,, since

h,,=plL=v~.J( lO,O00 m2/ha)

where

L=length of the laser transect (m).

Coefficients of variation (CV) tended to decrease approximately 5-15% in denser stands (85% canopy closure, Tileran and LaSelva-20 m) as the canopy became more spheric and as the denominator of the CVs, that is, average height, grew larger. In the simulated LaSelva-5 m canopy, with a canopy closure of 55%, coefficients of variation changed little/inconsistently with changes in canopy shape. Canopy closure estimates were completely unaffected by changes in the canopy shape assmnption since the simulated canopy covered the same area regardless of shape assumed above the base of the canopy.

Regression equations were calculated to predict seen basal area (h,), total basal area (b,), seen woody volume or biomass (v~), and total woody volume or biomass (v,)

330 Nelson

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Figure 1. A 100 m section of a "20-m-wide ground plot sampled at the LaSelva Biological Re- search Station near Puerto Viejo de Sarapiqui, CR. A) All tree canopies assumed to be cones; no noise (i.e., no random normal height variation) added. B) All tree canopies assumed to be half- ellipses; no noise added. C) All tree canopies assumed to be half-ellipses; noise added. The canopies have been modeled at a horizontal resolution of 1 m to reduce the density of canopy points displayed.

Modeling Forest Car, opy Heights: Effects of Canopy Shape 331

C.

40

30

g

10

^m, Tr'2' LoselVa ~2U

100~20 Ore' EIlip 5e

Figure 1. (Continued)

as linear functions of h,,, he, ca, and c~ for each canopy shape. The models were forced through the origin.

Table 3 reports the effects of different canopy shape assumptions on basal area, volume, and biomass estimates developed using airborne laser measurements. The airborne laser data acquired over the Tileran and La- Selva study sites were used to calculate the independent variables fi)r three sets of equations which predicted basal area, volume, and biomass as functions of h,, h~, c,, and c,.. For a particular stndy area (Tileran, LaSelva-5 m, LaSelva-20 m), the same segments of airborne laser data were processed using predictive equations developed under five different assumptions of canopy shape. Figure 2 illustrates the effects of canopy shape on estimates of total merchantable volume and total above-ground d u biomass. The tbllowing observations were noted:

• Mrborne laser estimates of basal area, volmne. and biomass and the standard errors of those estimates became larger as the canopy shape assumed in the simulator became more conic. On the average, the airborne laser estimates of total basal area assuming an elliptical crown were 2.8% larger than spherieal estimates, parabolic total basal area estimates were 6.7% larger than spherical estimates, and conical estimates were 14.3% larger than estimates developed assuming a spherical crown, regardless of regression/model approach. Considering total volume or biomass, elliptical estimates were 4.7% larger, parabolic

estimates were 1"2.0% larger, and conical estimates were 23.3% larger than estimates developed assuming a spherical canopy shape.

• A uniformly random distribution of the four canopy shapes produced estimates similar to parabolic or elliptical assumptions. Standard errors were likewise, in general, equivalent.

• The effects of the canopy shape assumption were strongly site-dependent in this study (Fig.'2). Tileran basal area and merchantable volume estimates were greatly affected (16.0% and 25.5%, respectively, across canopy shapes), while the LaSelva-20 m estimates of total basal area and total d U biomass exhibited almost no efl~ct (1.6% and -'2.0%, respectively across canopy shapes) as the canopy assumption changed.

Height variance as characterized by ¢, and c, ap- peared to drive this study site dependency. Simulated mean heights (both h(, and h~.) increased on all three study areas as simulated canopy shape was changed from cone to sphere, thereby decreasing the simple linear regression coefficients of mean height. It {}~llowed then that airborne laser estimates of basal area and vohnne or biomass decreased on all three study areas if these estimates were calculated as simple linear functions of /~,, or h,:. Likewise, the multiple linear regression coefficients of h, and h, decreased as expected. However, the additional interactions of the variance measures produced a relative increase in biomass in the LaSelva-90 m

332 Nelson

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Figure 2. Total merchantable volume (Tileran) and total above-ground dry biomass (LaSelva-5 m and LaSelva-20 m) estimated using airborne laser data. Five different assumptions of canopy shape-cone, parabola, ellipse, sphere, and a random assortment of these four shapes--were used to develop the equations which predict A) volume and B) biomass based on simulated tree height and height variability.

data set as canopy shape beeame more spheric. The small sample size used to calculate the LaSelva-20 m regressions (n=7) may be responsible for this insensitivity.

The assumption of a particular canopy shape, then, may not be trivial. An inappropriate assumption concerning canopy shape can introduce a bias in estimates of basal area, volume, or biomass, and this bias may be greater than 40% (Tileran volume, cone vs. sphere). The study site dependency suggests that the magnitude of the effect of the canopy shape assumption is very much affected by the spatial arrangement of the tree canopies. Different spatial arrangements will affect height variance measurements, especially ca and may overcome a ten- dency of airborne laser estimates of basal area, volume, or biomass to decrease as the assumed canopy shape be- comes more spheric. Ideally, the canopy shape of each

Modeling Forest Canopy Heights: Effects ~!f Canopy Shape 333

Table 3. Airborne Laser Esimates of Seen, Unseen, and Total Basal Area, Merchantable Woody Volume, and Above-Ground Dry, Biomass Calculated Using Five Different Assumptions of Canopy Shape ~

Airborne Laser Estinuzte

Cone Parabola Ellipse Sphere tlandom

Mean Std. Error Mean Std. Error Mean Std. Error Mean Std. Error Mean Std. Error

Tileran (u=8) BA, seen (me/ha) 26.57 3.51 23.68 3.07 21.71 2.84 20.68 2.54 23.59 2.611 BA, unseen (me/ha) 4.59 0.70 4.63 0.71 4.39 1t.75 3.91 /).89 4.26 I).95 BA, total (mZ/ha) 31.16 4.21 28.31 3.78 26.10 3.58 24.59 3.41 27.85 3.52 Mer. vohluw, seen (m3/ha) 236.34 47.48 199.59 46.19 175.33 44.83 156.78 44.98 203.66 40.52 Mer. vohune, unseen (m'~/ha) 26.29 6.06 27.21 5.69 2,5.84 5.80 22.37 6.50 23.23 7.70 Mer, volume, total (m~/ha) 262.63 53.31 226.80 51.84 201.18 50.55 179.15 50.86 226.89 47.35

LaSelva-5 m (n =611 BA, seen (mVha) 50.64 1.01 46.93 0.91 46.02 0.88 45.35 1t.89 45.34 tl.83 BA, unseen (m2/ha) 3.81 0.14 3.77 0.14 3.45 0.13 3.17 0.12 3.44 1).13 BA, total (nc'/ha) 54.45 1.13 50.70 1.03 49.47 0.99 48.52 0.99 48.77 0.93 Biomass seen (rot/ha) 339.39 8.51 310.02 7.71 295.17 7.45 279.19 7.17 295.89 6.67 Biomass, unseen (mr/ha) 12.51 0.54 12.71 0.59 11.18 0.54 9.74 0.46 11.29 0.49 Biomass, total (rot/ha) 351.90 9.02 322.73 8.26 306.35 7.95 288.93 7.611 307.18 7.11

LaSelva-20 m in = 131 BA, seen (me/ha) 34.44 1.34 33.41 1.31 33.57 1.34 33.68 1.49 33.65 1.34 BA, unseen (m2/ha) 4.30 0.60 4.01 0,59 3.82 0.60 3.56 0.51 3.77 0.63 BA, total (m~'/ha) 38.74 1.60 37.42 1.51 37.39 1.49 37.24 1.45 37.42 1.57 Biomass, seen (mr/ha) 200.08 8.46 194.17 8.68 190.84 8.90 197.87 9.1 I 197.27 9.49 Biomass, unseen (mr/ha) 17.28 2.32 16.39 2.26 16.02 2.40 17.97 2.13 16.33 2.44 Biomass, total (mr/ha) 217.35 10.08 210.57 10.22 206.85 10.55 215.84 10.45 213.60 1t).94

~' The airborne estimates utilize parametric multiple linear equations forced through the origin; 50 m segments o[" airborne laser data are considered, with 15 m gaps ]eR between adjacent segments.

t ree sampled along a ground t ransect would be no ted dur ing an inventory. Realistically, the canopy shape of an individual t ree may be difficult to ascertain in the field, especially in closed canopy situations where the general canopy lewq is 25 m or more over the observer 's head (e.g., a tropical moist fbrest). A more praetieal approach may be to note a general canopy shape for the forest in which the ground team is working, and then note ohvi- ous exceptions based on direct visual observation and/or based on kno,all growing habits of par t icular species.

CONCLUSIONS

The assumption of a par t icular canopy shape may have a marked affect on s imulated height measurements and on those regressions dew?loped to predic t basal area, volume, or biomass as a function of these height measurements. As canopy shapes progress from conic forms to

more spheric structures, canopy height increases, canopy height variation decreases (in closed canopy situations), and coefficients of variabili ty are stable or decrease. Since there is a direct relat ionship be tween average over- all height, canopy profile area, and canopy volmne (the three differ only by multiplicative constants), this work demonst ra tes that these three increase 8-10% when a parabol ic ra ther than a conic shape is assumed. An elliptic canopy is 16-18% t~dler than a conic tropical forest canopy, and a spheric eanopy is 23-'25% taller.

Parametr ic mult iple l inear regressions deve loped to predic t basal area, volume, and dDT biomass as fimctions of canopy height and height variability earl be, in some situations, profoundly affected. I f regressions are developed based on an assumption of a conic canopy shape,

and an airborne laser measures a canopy with similar total t ree height, which, however, exhibits spherically, shaped canopies, resultant est imates of basal area may be

overes t imated by 14% or more (on the average) and biomass or volume may be overes t imated 20% or more, on

the average. Since canopy shape may have a significant effect on

canopy height simulations, canopy shapes should he

no ted when field data are col lected for these purposes. Realizing that it is often difficult to view treetops in tall and/or dense forests, it is suggested that an in termedia te shape, for example, a parabolo id or an ellipsoid, he assumed, unless visual inspection or species growing habits suggest otherwise. An assumption of an in termedia te

shape or a random assor tment of shapes may mitigate potent ial ly large errors which might be incurred using a gener ic assumption of a cone or sphere.

R E F E R E N C E S

Aldred, A. H., and Bonner, G. M. 11985), Application of" airborne lasers to forest surveys, Intb. Rep. P1-X-51, Tech.

334 Nelson

Info. and Dist. Center, Petawawa National Forestry Inst., Chalk River, Ontario, 61p.

Maclean, G. A., and Krabill, W. B. (1986), Gross-merchantable timber volume estimation using an airborne LIDAR system. Can. J. Remote Sens. 12(1):%18.

Mawson, J. C., Thomas, J. w., and DeGraaf, R. M. (1976), PROGRAM HTVOL: the determination of tree crown volume by layers, USDA Forest Service Research Paper NE- 354, NEFES, Upper Darby, PA, 9 pp.

Nelson, R. (1994), The use of airborne laser altimetry to estimate tropical forest basal area, volume, and biomass, Ph.D. thesis, Virginia Polytechnic Institute and State University, Blacksburg, 145 pp.

Nelson, R., Oderwald, R., and Gregoire, T. G. (1997), Separat- ing the ground and airborne laser sampling phases to estimate tropical forest basal area, volume, and biomass. Re- nu)te Sens. Environ., this issue.

Nelson, R., and Yaya, J. (1990), Simulating forest stands to de-

rive laser profiling metrics, in Proc. lOth IGABSS, IEEE, Washington, DC, Vol. II, pp. 1221-1226.

Nelson, R. F., Krabill, W. B., and Maclean, G. A. (1984), l)e- termining lbrest canopy characteristics using airborne laser data. Renu~te Sens. Environ. 15:201-212.

Nelson, R. F., Krabill, W. B., and Tonelli, J. (1988a), Estimat- ing forest biomass and volmne using airborne laser data. Re- nu~te Sens. Environ. 24:247-267.

Nelson, R., Swift, R., and Krabill, W. (1988b), Using airborne lasers to estimate forest canopy and stand characteristics. J. For. 86(10):31-38.

Solodukhin, V. I., Zhukov, A. Ya., Mazhugin, I. N., Bokova, T. K., and Polezhai, V, M. (1977), Vozmozhnosti lazernoi aeros"emka profilei lesa (Possibilities of laser aerial photog- raphy of forest profiles). Lesn. Khoz. 10:53-58.

Solodukhin, V. I., Zheludov, A. V., Mazhugin, I. N., Bokova, T, K., and Shevchenko, K. V. (1985), The processing of laser profilograms for forest mensuratin. Lesn. Khoz. 12:35--37.

modeling forest canopy heights: the effects of canopy shape

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