An Expressionfor the Effectof Aspect,Slope,and
Habitat Type on Tree Growth
Note by A. R. Stage
Abstract. An expressionfor describingthe effect
of aspectand slope on tree growth is illustrated
by the relation of western white pine site index
to aspect,slope,and habitat type. Forest Sci. 22:
457-460.
Additional key words. Site, Pinus monticola,
western white pine.
STUDIES of the effect of environmental factors
on the growthof treesgenerallyassumethat
the effect of aspectcan be coded as a cosine
function with the minimum shifted to the
southwestquadrant and the maximum to the
northeastquadrant. This assumption,first
proposedby Gaiser (1951) with respectto
siteindex,hasbeenfoundto be an acceptable
(a-O) where B is the amplitude, a is the azimuth measured clockwise from north, and 0
is the phase shift that generally has been
assumed to be 45 ø (Trimble and Weitzman
1956). Supposenow that insteadof assuming
the optimum aspecta priori we wish to estimate the phase shift angle from the data at
hand. This can be accomplished
quite readily
by introducingboth the cosine of a and the
sine of a as two independentvariablesin the
multiple regressionfor predictingthe response
to aspect (e.g., equation (1) in Figure 1).
The regressioncoefficients estimated by least
squares for these two independent variables
can then be used to determine the phase shift
(0) and the amplitude (B) from the relations
given in Figure 1. The expressionsfor 0 in
degreesdepend on the signs of b• and b2 and
approximation in numerous later site index
studies. However, for other measuresof tree
the arc tan of the absolute value of the ratio:
growth such as periodic diameterincrement,
among the "formulae for reference" in appendices to many mathematical texts, but
their applicationto growthanalysesapparently
is not widely recognized.
The simple variables, sine and cosine of
azimuth, would suffice in the regressionif
all datawere obtainedfrom plotshavingabout
the same slope. However, plots on level
ground supply no information on the effect
of aspecton tree growth. Furthermore, it is
reasonableto assumethat the effect of aspect
will increaseon an adverseaspectup to the
the optimumaspectcan be quite different.
Indeed,in regression
models,the aspectvariable may have differenteffectsdependingon
the choice of other factors to be included in
the model. Beers and others (1966) have
shown how the phase of the cosinefunction
can be shifted to place the optimum at an
arbitrary aspect. Searching techniques to
locate the optimum for a given set of data as
used by Hartung and Lloyd (1969) can be
quite inefficient and dependfor their success
on a goodinitial guessof the location. Finally,
the useof higherorder sineor cosinefunctions
to represent asymmetrieswas introduced by
Carmean (1967).
The purposeof thisnote is to showhow the
foregoingdevelopmentscan be combinedinto
a single technique whereby the a priori assumption of a maximum in the northeast
quadrant (45 ø azimuth) and a minimum in
the southwestquadrantcan be replacedby
an empirically determined location of the
optimumwithout repeatedcalculationsof the
regressionfit. In addition, it is argued that
expressionsfor the effect of aspect should
always be consideredas terms involving an
interaction with slope. The simple symmetrmal cosine curve assumption can also be
modified to accommodatemore asymmetrical
forms of the responseto aspect.
The effect of aspectproposedby Beersand
others is stated mathematically as B cosine
[b2/b•l. Theseequations
arecommonly
found
angle of slope that is perpendicularto the
sun's rays. For greater slopes, the slight
advantageof a decreasein incoming radiation
is likely to be offset by decreasingsoil depth.
For these reasons, I recommend that the
variables to represent the combined effect of
slope and aspectbe defined as the tangent of
slope(slopepercent) times the sine and cosine,
respectively,of the azimuth. In this way,
plots on flat ground will have a zero value for
thesetwo variables,but plots on steepground
will have high weightsfor the sine and cosine
of aspect. In addition, the tangent of slope
Principal Mensurationist, USDA Forest Service, Intermountain Forest and Range Experiment
Station,Ogden,Utah 84401, stationedin Moscow,
Idaho, at the Forestry SciencesLaboratory, maintained in cooperation with the University of
Idaho. ManuscriptreceivedApril 15, 1976.
volume 22, number 4, 1976 / 457
shouldalsobe includedasa separateindependent variableto overcomethe symmetryof
the sine and cosinefunctions. Otherwise,the
steeperslopeson favorable aspectswould be
predictedto be more favorable than the flatter
Substitutingthese regressioncoefficients in
the expressions
givenin Figure 1, we find that
the data indicate a phase shift of arc tan
(0.08423/0.08070) = 46 ø. The amplitude of
the cosine function with the phase shift •s
slopes.The additivevariablefor slopepercent
¾0.08070"+
permits the model to describe adverse or
quently,the simplifiedpredictionequationis.
negligibleslopeeffectson the more favorable
aspectsand increasinglysevereconditionswith
•ncreasingslopes on the more unfavorable
Exampleof the applicationof thistechnique
•s taken from the study of the effect of environmentalfactorson the siteindex (S.I.) of
westernwhite pine (PinusmonticolaDougl.).
The regressionof site index on the factors of
aspect,slope, and habitat type (Daubenmire
and Daubenmire 1968) was calculatedto be:
0.08070s cos(a) + 0.08423s
sin(a) - 0.12634s + hab
where
ln(S.l.) = logarithmof siteindexfor
westernwhite pine
a = azimuth from north
s = slopepercent+ 100: tan (slopeangle)
4.16974 for Abies grandis/
Pachistimahabitattype
hab : 4.15770 for Tsuga/Pachistima
habitat type
4.41374 for Thuja/Pachistima
habitat type.
Conse-
ln(S.l.) • 0.11665s cos(a-46 ø) - 0.12634s
+ hab.
(slope angle) is -0.12634. It is oppositein
sign and of almost the same magnitude as the
amplitudeof cos (a-O) in the northeastquadrant. However, on the southwest facing
slopes,the slopeeffect accentuates
the adverse
aspecteffect. Figure 2 illustratesthe solution
of the final regressionequationfor two habitat
types. The plotting of the A bies grandis/
Pachistimahabitat type is omitted because•t
would nearly coincidewith the Tsuga/Pachistima habitat type.
In somecases,the symmetricalcosinefunction may not truly represent the effect of
aspect on tree growth. For example, the
function representedby the abovecoefficients
for site index of western white pine is illus-
trated by the solidline in Figure 3. Suppose
now that insteadof this relationshipan asymmetrical form, defined by the points representedby the filled circles,representsthe true
relationship.That is, the northeastand northwest aspects are equally favorable, and the
Relations for calculatingamplitude (B) and
phase shift (0) to expressthe effect of aspect
and slope on growth:
If:
a = azimuth in degreesfrom north
s: slope in percent + 100
then:
G = beq- Bs cos(a-O) q- b•
G = be+ b•scos(a) + ba sin(a) + b•s
(1)
where:
G:
0.11665.
Note that the regression coefficient for tan
aspects.
ln(S.l.):
0.08423":
(2)
where:
B = ¾ b•'•+ b••
growth response
and 0 is given by:
be: constantterm or sum of other predictor
effects in the regression
b2•• positive
negative
positivearctan
Ib2/bxl xs0
ø- arctan
negative
-arctan
Ib2/bxl xs0
ø+arctan
FIGURE1. Equationsfor calculatingamplitude(B) and phaseshiit (o) to
expressthe effect of aspecton growth.
458 / Forest Science
location
most adversesitesare on southwestexposures.
Such a relationshipcan be describedby using
additional sine and cosine terms for integer
multiples of the azimuth. That is, their arguments would be two, three, or perhaps four
t•mesthe azimuth. Such an approachwill be
familiar
to
those
who
have
used
of the extreme sites coincides sur-
prisinglywell with the commonlypostulated
45ø! However, for basal area growth of individual trees of the same specieson the same
habitattypes,the phaseshift was calculated
to be 201ø, indicating an optimum on the
southwesternaspectswithin the specified
habitat type. In fact, similar analysesfor
Fourier
analysisin engineering.Such an analysiswas
westernlarch, Douglas-fir, grand fir, western
redcedar, lodgepole pine, Engelmann spruce,
•ntroducedto forest mensurationistsby Ander-
son in 1937. When suchanalysiswas applied
to eight equally spaced data points from
F•gure 3, an expressionof the following form
and subalpinefir all showedsoutherlyaspects
to be superiorfor basalarea growth. Only
western hemlock, with an indicated optimum
was derived:
on the northeastaspects,coincidedwith the
In(S.L) ----0.0951s cos(a) + 0.0675s sin(a)
- 0.0142s cos(2a) q- 0.0126s
sin(2a) - 0.1263s + hab.
a prioriexpectation
(personalcommunication,
Robert A. Monserudand William R. Wykoff).
With the analysistechniquedescribedin this
note,theseunexpected
effectswere uncovered
usingregularmultipleregression
calculations
Predictionsusing this equation are shown by
the dashedline in Figure 3.
which are widely available.
In the western white pine example, the
SITE
84.00 -
80.75
77.50
74.25
SLOPEPERCENT
67.75
ß
64.50
-
Tsuga/pachistima
61.25
.•
•,oo I
0.00
I
I
I
I
I
I
I
45.00
90.00
135.00
180.00
225.00
270.00
315.00
360,00
DEGREES
FIGURE 2.
Trendsof westernwhitepinesiteindexwithslopeandaspectfor twohabitattypesin northern
ldaho,
volume 22, number 4, 1976
SITE
64.00,
-
///
E2.oo
''k
5•.00
.
//'/
•k
III/
//
60.00
59.00 -
58.00
57.00
55.00
0.00
I
I
I
I
I
I
I
45.00
90.00
135.00
180.00
225.00
270.00
315.00
360.00
DEGREES
FIGURE3. Comparison of symmetrical effect of aspect(solid line) with asymmetricaleffect (dashedline)
obtained by adding sine and cosineof twice the azimuth as additional regressionvariables.
Literature
Cited
ANDERSON,ROBERTT.
1937. The application
of Fourier's series in forest mensuration.
J For
35: 293 -299.
productivityresearch.J For 64:691-692.
CARMEAN,WILLA• H. 1967. Soil survey refinements for predicting black oak site quality
Soil Sci Soc Am
Proc
31:805-810.
DAUBENMIRE, R. R., and JEAN B. DAUBENMIRE.
1968. Forest vegetation of eastern Washing-
460 / Forest Science
GAMER,R.N.
1951. Relation between topog-
raphy, soil characteristics,and the site index of
white oak in southeastern Ohio.
BEERS,THOMASW., PETERE. DRESS,and LEE C.
WENSEL. 1966. Aspect transformation in site
in southeastern Ohio.
ton and northern Idaho. Wash Agric Exp Stn
Tech Bull 60, 104 p.
USDA
Forest
Serv, Cent StatesForest Exp Stn Tech Pap 121,
12p.
HARTUNG, ROBERT E., and WILLIAM J. LLOYD
1969. Influence of aspect on forests of the
Clarksville soils in Dent County, Missouri. J
For
67:178-182.
TRIMBLE, G. R., JR., and SIDNEY WEITZMAN
1956. Site index studiesof upland oaks in the
northern Appalachians. Forest Sci 2:162-173.