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.