Abstract. In many conceptual rainfall–runoff models, the water balance
differential equations are not explicitly formulated. These differential
equations are solved sequentially by splitting the equations into terms that
can be solved analytically with a technique called “operator splitting”. As
a result, only the solutions of the split equations are used to present the
different models. This article provides a methodology to make the governing
water balance equations of a bucket-type rainfall–runoff model explicit and
to solve them continuously. This is done by setting up a comprehensive
state-space representation of the model. By representing it in this way, the
operator splitting, which makes the structural analysis of the model more
complex, could be removed. In this state-space representation, the lag
functions (unit hydrographs), which are frequent in rainfall–runoff models
and make the resolution of the representation difficult, are first replaced
by a so-called “Nash cascade” and then solved with a robust numerical
integration technique. To illustrate this methodology, the GR4J model is
taken as an example. The substitution of the unit hydrographs with a Nash
cascade, even if it modifies the model behaviour when solved using operator
splitting, does not modify it when the state-space representation is solved
using an implicit integration technique. Indeed, the flow time series
simulated by the new representation of the model are very similar to those
simulated by the classic model. The use of a robust numerical technique that
approximates a continuous-time model also improves the lag parameter
consistency across time steps and provides a more time-consistent model with
time-independent parameters.
Impact of downscaled rainfall biases on projected runoff changes
