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 !