Abstract
The development of integrated water resources
management (IWRM) in the context of climate change and variability has
created the need for an extension of mathematical analyses of
hydrological system dynamics. The stationarity assumption of
hydrological time series, which has been widely used in the past,
cannot be further advocated. The classical uncertainty modelling
techniques based on probability theory cannot capture the multiple
facets of present hydrological uncertainties. The objective of this
study is to better capture the dynamics of the rainfall – runoff
process. To this end, this research develops a rainfall runoff modelling
approach that aims to capture the multiple sources and types of
uncertainty in a single framework. The main assumption is that
hydrological systems are non – linear dynamical systems which can be
described by stochastic differential equations (SDE). The dynamics of
the system is based on the Least Action Principle (LAP) as derived from
Noether’s theorem. The inflow process is considered as a sum of
deterministic and random components. The deterministic modelling of the
river discharge in the Ouémé river basin (Benin, West Africa), using the
hydrological model based on the least action principle (HyMoLAP),
revealed that this model is suitable to simulate the daily dynamics of
the river discharge. The stochastic formulation of HyMoLAP in terms of
SDE allowed to better take into account the dynamics of the process and
to explicitly show the proportion of the total variance of the discharge
that is attributable to each source of uncertainties in the rainfall –
runoff modelling. Then, the basic properties for the random component of
rainfall are considered and the triple relationship between the
structure of the inflowing rainfall, the corresponding SDE which
describes the river basin and the associated Fokker – Planck equations
(FPE) is analysed. The time – dependent probability distribution for the
resulting discharge is obtained in the form of fundamental and
approximate solutions of the FPE. A comparison is made between the time –
dependent probability distributions and the empirical distribution of
the outflow. The generalized FPE associated with the Langevin SDE
describing the river basin is derived in terms of the transition
probability distribution and characteristic function of the noise
generating process. This equation provides a useful tool for studying
the impact of various specific types of noises on the rainfall – runoff
process.