Should input rainfall dictate model timestep? - A ‘commensurability’ perspective
• Salim Goudarzi,
• David Milledge,
• Joseph Holden
Salim Goudarzi
Newcastle University

Corresponding Author:salim.goudarzi@newcastle.ac.uk

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David Milledge
Newcastle University
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Joseph Holden
University of Leeds
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## Abstract

Application of fixed timestep numerical schemes in engineering has long been criticized for their inaccuracy, inefficiency, and inconsistency across time-scales. Yet, to date, most hydrological models fix their timestep to the input rainfall resolution, instead of using adaptive schemes. Aside from their known maladies, we argue that fixed timestep schemes also suffer from ‘commensurability’ errors: errors that emerge when comparing quantities that are not precisely at the same spatial/temporal scales. At least at <= hourly resolutions, the observed discharge is a set of discrete measurements of an otherwise time-continuous (TC) quantity, but the modelled discharge is time-averaged (TA) across the fixed timestep. Hence the commensurability error when compared against one another during calibration. (In)significance of such errors simultaneously depends on the nonlinearity of the discharge within that timestep, and the timestep size. Consequently, these errors are the largest where they are potentially least acceptable to ignore, i.e., around peaks. Also, they tend to grow with timestep size (data resolution), unless timestep is detached from data resolution using adaptive schemes, which produce a TC solution. Importantly, since modern calibration procedures revolve around ‘fitting’ to observed discharge, such errors are likely undetectable in model’s curve-fitting performance, and instead are to be found in calibrated parameter-sets. Here, in a novel approach within the Generalize Likelihood Uncertainty Estimation (GLUE) framework with Limits of Acceptability (LOA) defined a-priori, and for a micro-catchment case study, we calibrate a TA and a TC version of Dynamic-TOPMODEL to datasets at different resolutions. Through experimentation with the calibrated parameter-sets, we estimate the relative (to TC version) magnitude of the time-commensurability errors resulting from fixing the timestep to input rainfall. Our findings confirm the overall insufficient accuracy, inefficiency of timestepping, and inconsistency across resolution when fixing the timestep. We find that for calibration data resolution >10min, time-commensurability errors become very significant.