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Modelling combined effects of elevation, aspect and slope on species-presence and growth. Albert R. Stage and Christian Salas. Old ideas. French scientists modelled wine cork lengths on different sides of oak trees 50 years ago with: a · Cos(aspect) + b ·Sin(aspect) - PowerPoint PPT Presentation
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Modelling combined effects of Modelling combined effects of elevation, aspect and slope elevation, aspect and slope on species-presence and on species-presence and
growthgrowth
Albert R. StageAlbert R. Stage
andand
Christian SalasChristian Salas
Old ideasOld ideas
• French scientists modelled wine cork lengths on different sides of oak trees 50 years ago with:
a·Cos(aspect) + b·Sin(aspect)
• Beers, Dress and Wensel 40 years ago (1966) recommended a·Cos(aspect + phase shift)
where phase shift for the adverse aspect was assumed to be = SW
• Stage 30 years ago (1976) added an interaction with slope to represent white pine site index:
slope·[a·Cos(aspect) + b·Sin(aspect)+ c] and thereby allowing the data to determine the phase shift.
Trig TricksTrig Tricks
• Stage(1976) is a generalization of Beers, Dress and Wensel (1966) because:
y = b0 + b1s + b2·s·cos(α) + b3·s·sin(α)
is identical to:
y = b0 + b1·s + cos(α - β)
for β = +arctan(b3/b2) if b2 > 0 or −arctan(b2/b3) if b3 >0.
bb2
3
2
2
Now what about Elevation? Now what about Elevation?
• Roise and Betters (1981) argued that optimum phase shift reverses between elevation extremes-- but omitted aspect/slope relations in their formulation.
• Here we combine these concepts in terms of main effects of elevation with two elevation functions interacting with slope/aspect triplets.
Introducing the two Introducing the two elevation/aspect interactions:elevation/aspect interactions:
• Behavior:– Sensitivity to elevation increases toward the
extremes (contra Roise and Betters 1981)– Scale invariant– Linear model preferred
Introducing the two Introducing the two elevation/aspect interactions:elevation/aspect interactions:
F1(elev)·slope·[a1·Cos(aspect) + b1Sin(aspect)+ c1] +
F2(elev)·slope·[a2·Cos(aspect) + b2·Sin(aspect)+ c2]
+ d1·F3(elev)
Some alternative pairs of functions:F1 (low)
Constant = 1
elevation
Log(elevation)
Log (k·elevation)=Log (elev) + log(k)
F2 (high)Square of elevation
Square of elevation
Square of elevation
Square of elevation
Challenging hypothesis with DATA! Challenging hypothesis with DATA!
• Where there is agreement---
Classifying forest/non-forest in UtahClassifying forest/non-forest in UtahSlope = 20%Slope = 20%
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 90 180 270 360
Aspect
Dis
cri
min
an
t
Non-f
ore
st >
|< F
ore
st
Elev. =1750 m.
Elev. = 4000 m.
Utah productivityUtah productivity
MAI - UtahSlope =25%
0
10
20
30
40
50
60
1.5 2 2.5 3 3.5 4 5 6
Elevation (m/1000)
Cu
. F
t.
north
south
level
Challenging hypothesis with DATA! Challenging hypothesis with DATA!
• Where there is agreement---
• And where there is not !
Douglas-fir Height GrowthDouglas-fir Height GrowthF1 = ln(elev), KF1 = ln(elev), K F2 = elevF2 = elev22
30
35
40
45
50
55
N E S W N
Aspect
As
ym
pto
te (
m)
853 m.
1219 m.
1768 m.
30
35
40
45
50
55
N E S W N
Aspect
As
ym
pto
te (
m)
640-1082 m.
1083-1311 m.
1312- 2073 m.
Douglas-fir Height GrowthDouglas-fir Height Growth3 elevation classes3 elevation classes
Not an artifact !
So So · · · · · · ??
• Proposed formulation consistent with ecological hypotheses concerning elevation-aspect-slope relations · · ·
• But allows data to define some surprises !