Set-membership approach for identification of parameter and prediction uncertainty in power-law relationships: The case of sediment yield

Power laws are used to describe a large variety of natural and industrial phenomena. Consequently, they are used in a wide range of scientific research and management applications. This paper focuses on the identification of bounds on the parameter and prediction uncertainty in a power-law relation from experimental data, assuming known bounds on the error between model output and observations. The prediction uncertainty bounds can subsequently be used as constraints, for example in optimisation and scenario studies. The set-membership approach involves identification and removal of outliers, estimation of the feasible parameter set, evaluation of the feasible model-output set and tuning of the specified bounds on model-output error. As an example the procedure is applied to data of scattered sediment yield versus catchment area (Wasson, 1994). The key result is an un-falsified relationship between sediment yield and catchment area with uncertainty bounds on its parameters. The set-membership results are compared with the results from a conventional least-squares approach with first-order variance propagation, assuming a zero-mean, symmetrical error distribution.