Transcript Modelling combined effects of elevation, aspect and slope on species-presence and growth Albert R.

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)
• 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 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 + b  b cos(α - β)
for β = +arctan(b3/b2) if b2 > 0 or
−arctan(b2/b3) if b3 >0.
2
2
2
3
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
elevation/aspect interactions:
• Behavior:
– Sensitivity to elevation increases toward the
extremes (contra Roise and Betters 1981)
– Scale invariant
– Linear model preferred
Introducing the two
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)
F2 (high)
Constant = 1
Square of elevation
elevation
Square of elevation
Log(elevation)
Square of elevation
Log (k·elevation)=
Log (elev) + log(k)
Square of elevation
Challenging hypothesis with DATA!
• Where there is agreement---
Classifying forest/non-forest in
Utah
Slope = 20%
0.15
Elev. =1750 m.
Discriminant
Non-forest >|< Forest
0.1
Elev. = 4000 m.
0.05
0
-0.05
-0.1
-0.15
-0.2
0
90
180
Aspect
270
360
Utah productivity
MAI - Utah
Slope =25%
60
Cu. Ft.
50
40
north
south
level
30
20
10
0
1.5
2
2.5
3
3.5
Elevation (m/1000)
4
5
6
Challenging hypothesis with DATA!
• Where there is agreement--• And where there is not !
Douglas-fir Height Growth
F1 = ln(elev), K F2 = elev2
55
853 m.
1219 m.
Asymptote (m)
50
1768 m.
45
40
35
30
N
E
S
Aspect
W
N
Douglas-fir Height Growth
3 elevation classes
55
640-1082 m.
1083-1311 m.
Asymptote (m)
50
1312- 2073 m.
45
40
Not an artifact !
35
30
N
E
S
Aspect
W
N
So · · · ?
• Proposed formulation consistent with
ecological hypotheses concerning
elevation-aspect-slope relations · · ·
• But allows data to define some surprises !