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Robert H. Knapp, Jr. and Richard L. Williamson
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Forest Sci., Vol. 30, No.2, 1984, pp. 284-290
Copyright 1984, by the Society of American Foresters
0
ABSTRACT. Silviculturists, forest micrometeorologists, and hydrologists have interest in shade
cast by tree crowns because of its influence on soil surface :temperatures and on air temperatures
near the ground. We provide equations which describe the boundary of illumination on sunlit
crowns and estimate the amount of shade cast by tree crowns when one assumes that crown
shapes can be represented by paraboloids, cones, or ellipsoids. FoREST Sci. 30:284-290.
ADDITIONAL KEY WORDS.
Shading (seedlings), shade, tree crowns, insolation, microclimatology.
SHADE CAST BY TREE CROWNS is of interest to silviculturists, forest micrometeorologists,
and hydrologists because of its influence on soil surface temperatures and on air temper­
atures near the ground (Halverson and Smith 1979). Temperature near the soil-air interface
is a critical factor in seedling survival when regenerating severe sites. Shade affects rate of
snowmelt and hence the duration and rate of runoff from forest watersheds (Alexander
and Watkins 1977). Shade affects the period of access to roads and campgrounds located
in snow zones. Shade affects drying rates and therefore fire hazards.
The forester controls the amount of shade cast by tree crowns through manipulation of
stand density in thinning and regeneration cuts. There are many references in forestry
literature to the importance of shade in establishment and early survival of regeneration
on severe sites. For example, Korstian ( 1925), Isaac ( 1943), and Williamson ( 1973) agree
that about 50-percent canopy density is optimum for establishment of regeneration of
Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) on severe sites.
How does one specify the residual stand required (in, for example, a shelterwoo'U'regen­
eration cut) to attain a desired degree of shading?
Given the relationship between crown diameter and tree size, one could estimate the
vertical projections of tree crowns for any given residual stand. But, in northern latitudes
the sun's rays are not vertical, and for long-crowned trees the actual shadow area may
differ considerably from the vertical projection of the tree crown.
Some information on shade cast by trees is available from reports on shadow patterns
of stand borders within clearcuts (Halverson and Smith 1974, Gordon 1968), but these
results mainly concern shadow length. We could find no information on shadow areas of
individual trees.
Shapes of some tree crowns, such as those of true firs and most young-growth conifers,
are fairly regular and can be approximated as solids of revolution generated by conic
sections. For instance, Curtis and Reukema ( 1970) assumed the crowns of 45-year-old
Douglas-fir trees to be paraboloids. One could assume a conical shape for most true fir
crowns.
Given the equation of the appropriate solid of revolution and the elevation angle of the
sun's rays, one can calculate a second equation for the locus of points where a ray just
grazes the crown. These tangent rays, projected to ground level, mark out the crown's
shadow, for which a third equation (our principal result) is a simple extension of the second
calculation.
.,,
SHADow BouNDARIES
The points on the crown where the sun's rays are tangent form the boundary between the
illuminated and shadowed portions. A tangent ray must satisfy three conditions: ( 1) it must
61Ym=,,_
The authors are, respectively, Member of the Faculty (Physics), The Evergreen State College,
pia, Washington 98505; and Silviculturist, U SDA Forest Service, Pacific Northwest Forest and Range
Experiment Station, Olympia, Washington 98502. Manuscript received 4 January 1982.
284 /FoREST SCIENCE
shadowed
portion of
crown
h
b
�---z cot 9 --�
R'
2(h·b)
tan 9
I. Detail of relation between points in crown shadow boundary and points in ground shadow
boundary.
FIGURE
pass through a point (x,y, z) lying on the crown's surface; (2) it must lie in the plane tangent
to the surface at this point; and (3) it must point in the direction of the sun (See Fig. 1).
In the Appendix, we derive a general formula for the shadow boundary:
x = R2/2*tan 8*( -df/dz)
(A4)
where the functionf( z) defines the particular surface under study, and 8 is the sun's altitude.
Resulting shadow boundary equations for paraboloid, cone, and ellipsoid are
= - l l(h - b): x = R2*tan 8/[2*(h - b)];
Cone df!dz = -2*( h - z)l( h - b)2: x = R2*tan 8*(h - z)l( h - b)2; and
Ellipsoid df!dz = -2*(z - z0)/[(h - z0)2 - (b- z0)2]:
x = R2*tan 8*(z - z0)/[( h - z0)2 - (b - z0)2].
Paraboloid df!dz
( l a)
( 1b)
( l c)
These describe the boundaries one would see in looking at the crown along the y-axis;
i.e., along the horizontal direction perpendicular to the sun's rays. Surprisingly, these are
all straight lines: vertical for the paraboloid; for the cone, sloping negatively down from a
z-intercept at the crown height h; for the ellipsoid, the third line sloping positively, and
cutting the z-axis at z0, the height of the ellipsoid's equator. The strong qualitative difference
in slopes means that, theoretically, one could qualitatively judge the crown's shape simply
by looking at the shadow boundary from the side and noting whether it sloped towal'�he
crown or away from it, or was vertical (see Fig. 2). Similarly strong differences would appear,,
in vertical views: the paraboloid would show a straight boundary perpendicular to the sun's
azimuth; the cone would show a straight-sided wedge with its apex toward the sun; the
VOLUME 30, NUMBER 2, 1984 /285
+x
-X
+X
+X
cone
paraboloid
FIGURE
2.
-X
ellipsoid
Shadow boundaries for paraboloidal, conical, and ellipsoidal crowns.
ellipsoid would show a crescent shape with apex away from the sun. Our experience has
been that vertical views of young Douglas-fri stands do show shadow boundaries of conical
and paraboloidal shapes. In our judgement, these qualitative features are best observed on
negatives of black-and-white vertical aerial photographs of the stand in question, with scale
of photography about 1:2,400. This allows one's point of view to be removed far enough
from the trees so that irregularities are smoothed out.
AREA OF HoRIZONTAL SHADOW CAsT
BY
A CRoWN
The rays that graze the crown at any point (x, y, z) on the shadow boundary pass on to
strike the ground at a corresponding point (x', y', 0), marking out the boundary of the
horizontal shadow. The diagram (Fig. 3) shows how to fn
i d the coordinates of the corre­
sponding point, called P' in the diagrams, given the position of the initial point P?the y
coordinates of both points are the same; x', the x coordinate of P', is the sum of two
lengths-the horizontal leg of the right triangle PP'G and the displacement of the shadow
boundary from the tree axis; the z coordinate of P' is, of course, zero. In other words,
z'
=
0; y'
=
y; x'= x + z cot 8.
(2)
One can fn
i d the equation of the ground shadow boundary by solving the equation of the
TABLE 1. Equations for calculating areas of horizontal shadows of truncated conic sec­
tions of revolution.
Paraboloid: intercept parameter a = R*tan /1/[2*(h- b)]
(l A ) (a:,; 1)• A= (2/ 3)*R2*(1- a2)312 /a + 1r*R2*(l-arc cos a)h +R2*a (l-a2)112
(lB) (a > 1) A= 1r*R2
Cone: intercept parameter (3 = R*tan /1/(h- b)
(2A) ((3:,; 1) A= R2*(1- B2)112 / B + 1r*R2* (l -arc cos (3)17r
(2B) ((3 > 1) A= 1r*R2
Ellipsoid: intercept parameter y = (](2/R)*tan O*(b- z0)
(3A) (y:,; 1) A= 1r*K2*(h- z0)2*(1 +cot2/1/K2)112*(1/2 +arc sin "Ah) +"A* (l- t-2)112
+ 1r*R2*[(l -arc cos "A/1r)- y*(l - y2)112/1r] where ](2 = R2/[(h- z0)2- (z0- b)2] A= (z0- b)l(h
z0)* (](2*tan211 + 1)112
(3B) (y > 1) A= 1r*](2*(h- z0)2*(1 +cot /1/](2)112
• There are two cases for each conic, depending on the value of an "intercept parameter."
h = height of vertex; b = height of circular base; R = radius of base; z0 = height of ellipsoid's equator;
/1 sun elevation. See A ppendix for further discussion.
=
286 I FOREST SCIENCE
z
2
R
2{h-6)
h
b
parabolic
shadow
X
FIGURE 3.
y
Relation between crown shadow and ground shadow for paraboloidal crown.
surface (A l ), the equation of the crown shadow boundary ( 1) and the projection equation
(2) for y' in terms of x'. For our three conic sections, the results are in Table A2 of the
Appendix.
We are dealing with truncated conic sections with circular bases. The crown shadow
boundary will usually intersect the base-in fact only for a more than half complete ellipsoid
at high sun elevations will it fail to do this. The sunward (south) end of the shadow is cast
by the portion of the base on the sunward side of the shadow boundary, and is in fact a
portion of a circle of the same radius.
The area of the shadow, then, is the sum of two areas (circular and projected conic
portions). As indicated in Figure 3 the total area can be found by straightforward, though
tedious, integration. The results are in Table 1.
Each of these area formulas contains a dimensionless "intercept parameter" (a, (3, and
f', respectively) that governs the fraction of the ground shadow taken by the circular-t;J ion
due to the circular base of the crown. Larger sun altitudes correspond to larger values·of
all of these parameters, and in each case the critical value 1 marks a qualitative change in
the shadow-to a pure circle for paraboloids and cones, and to a pure ellipse for ellipsoids.
VOLUME 30, NUMBER 2, 1984 /287
TABLE 2. Shadow area predictions for trees with conoidal crowns, for three sun altitudes
({)), and for three degrees of slope (o), with aspect 220 degrees, in square meters.
Slope
Shadow area by crown shape
Sun altitude
0
20
40
Paraboloid • Conoidb
'
Ellipsoidc
30
474
370
543
55
230
188
266
80
119
113
160
30
320
250
366
55
192
157
223
80
113
108
153
30
224
175
257
55
158
129
183
80
107
102
144
a,b,c Live-crown length (h
b)
a,b Radius of crown at base (R)
c Radius of crown at base (R)
-
30 m.
=
=
=
6 m.
5.39 m.
SHADOW AREAS ON OTHER SLOPES AND ASPECTS AND FOR OTHER SUN DIRECTIONS
Thus far, we have assumed that sun azimuth is due south and ground surface is horizontal.
For other times of day and for all slopes and aspects, one can calculate the relative change
in area (horizontal basis) as follows (all angles and directions in degrees):
1. Assume the x andy axes are rotated so that the x axis points toward the sun's azimuth.
2. Obtain the sun's declination (d) from any one of numerous ephemerides or surveying
;.aids.
3. Determine altitude of the sun ({)) by: sin {)= sin L sin d + cos L cos d cos h, where
L= latitude of observer, and h= hour angle of sun, angular displacement from due south,
usually assigned 15° per hour.
4. Determine slope perpendicular to contours (o). Assuming the sun is due south, assign
positive values to northerly slopes and negative values to southerly ones.
5. Determine the difference in direction (o) between aspect and azimuth to the sun. No
need to assign positive or negative signs.
6. Relative change in shade area then equals: ala(in percent)= 100*sin(o)*sin(90 {))*cos(o)/sin({) - u).
.
.
where a= area of shade cast on a horizontal surface. Example, for 10 AM, with the sun's
declination 10°N, latitude 45°N, slope 24 degrees and southerly with respect to the sun,
and the angular deviation between directions for slope and sun of 25 degrees.
sin 45° sin 10° + cos 45° cos 10° cos 30°= 46.54°, -24 degrees, 25 degrees, 30 degrees. {)
u
o
h
Then,
a (in percent)= 100*sin(180 - ( -24))*sin(90 - 46.54)*cos 25/sin(46.54 - ( -24))
where {)= sun's altitude, and h = hour angle of the sun, angular displacement from assumed
due south, usually assigned 15° per hour. Note that all negative signs must be adhered
to. Results in terms of square meters of sh de under various conditions are illustrated in
Table 2.
CoNCLUSIONS
We have presented equations which describe the shadows cast by tree crowns hose shapes
are approximated by paraboloids, cones, or ellipsoids, when the crown is fully exposed to
288 /FOREST SCIENCE
----\.----.
the sun and radius of crown (R), height to tip (h), and height to base of crown (b) are
known.
Height to tip is a routine forest tree measurement. Height to base of live crown and
crown radius are less commonly measured, but can be measured directly or estimated from
other tree and stand characteristics (Beekhuis 1965, Curtis and,,Reukema 1970, Kramer
1966, Smith and Bailey 1964).
The equations given do not allow for overlap among tree shadows. Therefore, a simple
summation of estimated shadow areas over a given lanq area provides a reasonable estimate
of relative amount of shade only in relatively open stands, and near noon and near the
summer solstice, conditions where mutual shading of crowns is at a minimum. In shelter­
wood regeneration cuttings, overwood density is usually low enough so that shadow overlap
is minimal, particularly during the midday period in summer which is the critical period
for insolation-caused mortality. Therefore, it seems reasonable to use such calculated total
shadow areas as a measure of shading under such conditions, and a basis for specifying
number of residual trees required to achieve a desired degree of shading.
These equations may also have potential usefulness as a component of further mathe­
matical development to estimate amount of shade and interception of solar energy when
shadow overlap can be expected.
LITERATURE CITED
ALEXANDER, R. R., and R. K. WATKINS.
1977. The Fraser Experimental Forest, Colorado. USDA
Forest Serv Gen Tech Rep RM-40, 32 p. Rocky Mt Forest and Range Exp Stn, Ft. Collins, Colo.
BEEKHUIS, J.
1965. Crown depth of radiata pine in relation to stand density and height. N Z
J
For
10(1):43-61.
CuRTIS, R. 0., and D. L. REUKEMA. 1970.
Crown development and site estimates in a Douglas-fir
plantation spacing test. Forest Sci 16(3):287-301.
GoRDON, D. 0. 1968.
Tree shadow patterns and illumination measurements within clearcut strips
and irregular openings in a true fir forest. USDA Forest Serv Res Note PSW-172, 9 p. Pac
.
')
)'
Southwest Forest and Range Exp Stn, Berkeley, Calif.
HALVERSON, H. G., and J. L. SMITH.
;
·
,
1974. Controlling solar light and heat in a forest by managing
shadow-sources. USDA Forest Serv Res Pap PSW-102, 14 p. Pac Southwest Forest and Range
Exp Stn, Berkeley, Calif.
HALVERSON, H. G., and J. L. SMITH.
1979.
Solar radiation as a forest management tool: a primer
of principles and application. USDA Forest Serv Gen Tech Rep PSW-33, 13 p. Pac Southwest
Forest and Range Exp Stn, Berkeley, Calif.
IsAAc, L. A. 1943. Reproductive habits of Douglas-fir. Charles Lathrop Pack Forestry Publication
for the U.S. Forest Service. 107 p.
KoRSTIAN, C. F.
1925.
Some ecological effects of shading coniferous nursery stock. Ecology VI( l ):
48-51.
KRAMER, H. 1966. Crown development in conifer stands in Scotland as influenced by initial spacing
and subsequent thinning treatment. Forestry 39(1):40-58.
SMITH, J. H. G., and G. R. BAILEY.
1964.
Influence of stocking and stand density on crown widths
of Douglas-fir and lodgepole pine. Commonwealth For Review 43(3):243-246.
WILLIAMSON, R. L. 1973. Results of shelterwood harvesting of Douglas-fir in the Cascades of western
Oregon. USDA Forest Serv Res Pap PNW-161, 13 p. Pac Northwest Forest and Range Exp Stn,
Portland, Oreg.
APPENDIX: CALCULATION OF SHADOW BOUNDARIES
We are concerned with three surfaces of revolution: paraboloid, cone, and ellipsoid. The
vertex of each will be at a height h, and each will be truncated to have a circular base of
radius R at a height b. We take the origin of rectangular coordinates at the base of the tree,
,•with the z-axis vertical and the x-axis pointing in the sun's direction (Fig. 3).
h
Table Al shows that each surface has an equation of the form
x2 + y2
= R 2j{z)
(Al)
VOLUME 30, NUMBER 2, 1984 /289
Assuming that the sun is at an altitude angle () and its rays are all parallel, the unit vector
in the sun's direction has coordinates
n
=
( -cos e, 0, sin ()), (A2)
The tangent plane for a general surface of three variables, F(x, y,, z) 0, is defined by the
condition that any vector n lying in the surface must be perpendicular to the gradient of
the function f
=
(A3)
n·'VF=O where \1F is the vector (aF/ax, aF/aY, aF/az) and the dot indicates the scalar product of
two vectors.
Here, F(x,y,z) x2 + y2 - R2f(z), so we have
=
or,
x
=
(R 2/2)*tan ()*( -df! dz).
(A4)
Inserting explicit forms for f(z) give the shadow boundary results in equation (1) of the
main text.
TABLE AI.
x2 + y>
x2 + y>
X2 + y>
Paraboloid:
Cone:
Ellipsoid:
=
=
•
R2*(h- z)/(h- b)
R2*(h- z)'!(h- b)2
R2*[(h- Zo)'-(z- Zo)']l[(h- Zo)2-(b-Zo)'J
• h height of vertex; b height of circular base; R radius of base; z0
height of ellipsoid's
equator. The ellipsoid formula is slightly rearranged from the standard form (x2 + y)/a2 +r;2/b2 1
to incorporate parameters convenient for this discussion. Measuring the radius at some convenient
point of the crown at a height z (we recommend the point three-quarters of the way up the live crown
for ease of observation) provides the following values to substitute in the formula: (x,y,z)
(r,O,z).
The formula can then be solved for z0, yielding:
=
=
r
=
Swfaces of revolution for conic sections.
=
=
=
=
z0
=
TABLE A2.
(112)*{(r/R)2*[(h2- b2)-(h2- z2))/(r/R)2*[(h- b)-(h- z)j}.
Equations for crown shadow boundaries projected to the horizontal.
[ These equations describe only the shadows of conic solids of revolution. The truncated solids
encountered in practice are handled as described in the main text.]
Paraboloid: x'
where Cp
Cone:
x
where Cc
Ellipsoid:
=
=
=
[ lf2CP(l- l /2CP)tan IJ + hcPcot IJ]-y'2cot IJ
R2/(h- b). This is the equation of a parabola.
=
h*cot IJ- [y'[*[(l- C/tan21J)/(C/tan21J)jl12
R2/(h- b)2• This is the equation of two straight lines.
(X- z0*cot IJ)2[(C/tan21J + l)/(C/tan21J + Ce)J + y12
=
Ce*(h- z0)'
where ce R2/[(h- Zo)2-(b- Zo)'J. This is the equation of an ellipse displaced from
the origin.
=
290 /FOREST SCIENCE
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(