Exploring equifinality in a landscape evolution model

Description/Abstract

Model equifinality is the property by which very similar model outputs can be generated by
many different combinations of model inputs. It is known in numerical models used in other
disciplines, and is thought to be likely in landscape evolution models (“LEMs”) also, as they
incorporate many process parameters of uncertain value. LEM equifinality, if pervasive,
would be a serious obstacle to falsifying working hypotheses and would frustrate landscape
evolution research, but to date it has not been quantified. This is attempted here, by
sampling a LEM’s response in its parameter space. A well known LEM (‘GOLEM’, Tucker
& Slingerland, 1994), used here as an exemplar, is applied to evolution of a c. 38 km2, 4th
order catchment in the Oregon Coast Range. Ten of GOLEM’s parameters are selected for
variation, covering mass movement, channel formation, fluvial erosion and weathering
processes, and value ranges appropriate for the catchment are established from published
data and calibration. Parameter space sampling is then carried out using a response surface
methodology approach which reduces by c. 3 orders of magnitude the simulation run size
needed to explore the 10-D parameter space. Initial simulations are run sampling the space
according to a central composite design of 149 targeted parameter value combinations,
which afford estimation of all parameter main and two-way interaction effects. Model
outputs at 100,000 years are summarised by four metrics (sediment yield, drainage density,
sediment delivery ratio, and a topographic metric), which serve as landscape descriptors.
Equations, or “metamodels”, are derived by regression to describe each metric as a function
of the GOLEM parameters, and further simulations allow testing and improvement of model
fits (R2 of c. 98% for the sediment yield, drainage density and sediment delivery ratio, and c.
92% for the topographic metric). The parameter space is then sampled rapidly and densely
(>>106 times), using each metamodel to predict GOLEM’s output at each sample point.
Results are compared with a reference value for each metric, to obtain equifinal proportions
in a range of permitted tolerance bands around the reference, and using a bootstrap to aid
calculation of confidence intervals. The likelihood of obtaining an equifinal result is found
to depend on the tolerance band and the metric e.g. the equifinal probabilities for drainage
density are estimated to be c. 26% and 58% respectively in the 2% and 5% tolerance bands,
compared with c. 68% and 99% for the sediment delivery ratio in the same bands. Where
combinations of metrics are used, the polymetric equifinal probability is often lower (and
never higher) than it would be for any of the component metrics used singly. Also, the
equifinal probability for any metric and tolerance band usually decreases as the number of
parameters employed in the model increases. More generally, equifinal probabilities are
seen to result from the combinations of parameter main effects and interactions driving each
metric, thus allowing equifinality to be explored through the use of metamodel archetypes.
Further research using other LEMs is needed, and the response surface methodology is
recommended for both its computational efficiency and clarity in this respect.