Exploring equifinality in a landscape evolution model

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.

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Identifiers

ORCID for Stephen E. Darby: orcid.org/0000-0001-8778-4394

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