1. Modelling combined effects of elevation, aspect and slope on species-presence and growth Albert R. Stage and Christian Salas

2. 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.

3. Trig Tricks Stage(1976) is a generalization of Beers, Dress and Wensel (1966) because: y = b 0 + b 1 s + b 2· s·cos(α) + b 3· s·sin(α) is identical to: y = b 0 + b 1· s + cos(α - β) for β = +arctan(b 3 /b 2 ) if b 2 > 0 or −arctan(b 2 /b 3 ) if b 3 >0.

4. 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.

5. Introducing the two elevation/aspect interactions: Behavior: –Sensitivity to elevation increases toward the extremes (contra Roise and Betters 1981) –Scale invariant –Linear model preferred

6. Introducing the two elevation/aspect interactions: F 1 (elev)·slope·[a 1 ·Cos(aspect) + b 1 Sin(aspect)+ c 1 ] + F 2 (elev)·slope·[a 2 ·Cos(aspect) + b 2 ·Sin(aspect)+ c 2 ] + d 1 ·F 3 (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

7. Challenging hypothesis with DATA! Where there is agreement---

8. Classifying forest/non-forest in Utah Slope = 20%

9. Utah productivity

10. Challenging hypothesis with DATA! Where there is agreement--- And where there is not !

11. Douglas-fir Height Growth F1 = ln(elev), K F2 = elev 2

12. Douglas-fir Height Growth 3 elevation classes Not an artifact !

13. So · · · ? Proposed formulation consistent with ecological hypotheses concerning elevation-aspect-slope relations · · · But allows data to define some surprises !