Use this URL to cite or link to this record in EThOS:
Title: Application of fractional calculus to rainfall-streamflow modelling
Author: Borthwick, Martin Francis
ISNI:       0000 0004 2691 0782
Awarding Body: University of Plymouth
Current Institution: University of Plymouth
Date of Award: 2010
Availability of Full Text:
Access from EThOS:
Access from Institution:
There is evidence that hydrologic systems exhibit memory processes that may be represented by fractional order systems. A new theory is developed in this work that generalises the classical unit hydrograph technique for the rainfall-runoff transformation. The theory is based upon a fractional order linear deterministic systems approach subject to an initial condition and is taken to apply to the entire rainfallstreamflow transformation (i.e. including baseflow). The general equation for a cascade of time-lagged linear reservoirs of fractional order subject to a constant initialisation function is derived, and is shown to be a form of fractional relaxation model. Dooge's (1959) general theory of the instantaneous unit hydrograph is shown to fit within the new theoretical framework. Similarly the relationship to the general storage equation of Chow and Kulandaiswamy (1971) is demonstrated. It is shown that the correct initialisation of cascade models requires a substantial number of initial conditions which may limit the viability of applying them in practice. Consequently, the differential formulation of the classical Nash cascade has been corrected and reinterpreted. The unbounded nature of the solution to the convolution integral form of the single fractional relaxation model is overcome by application of the Laplace transform of the pulse rainfall hyetograph following Wang and Wu (1983). The model parameters are fitted using the genetic algorithm. The fractional order cascade equations are tested for classical rainfall-runoff modelling using a set of 22 events for the River Nenagh. The cascade of 2 unequal fractionalorder reservoirs is shown to converge to that of the integer order case, whilst the cascade of equal reservoirs shows some differences. For the modelling of the total rainfall-streamflow process the single fractional order reservoir model with a constant initialisation function is tested on a selection of events for a range of UK catchment scales (22km^ to 510km ). A rainfall loss model is incorporated to account for infiltration and evapotranspiration. The results show that the new approach is viable for modelling the rainfall-streamflow transformation at the lumped catchment scale, although the parameter values are not constant for a given catchment. Further work is recommended on determining the nature of the initialisation function using field studies to improve the identification of the model parameters on an event-by-event basis.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available