Assessing the TOPKAPI non-linear reservoir cascade approximation by means of a characteristic lines solution

TOPKAPI is a physically based distributed rainfall-runoff model derived upon the assumption that the horizontal flow at a point in the soil, over the slopes and in the channel network can be approximated by means of a kinematic wave model. The TOPKAPI model combines this kinematic approach with a digital elevation model (DEM)-based description of a basin. Similarly to what is done in the finite element approach, the basic model equations are derived by integrating in space the point process equations up to a finite dimension, the pixel, thus converting the original kinematic partial differential equations into a cascade of finite dimension non-linear reservoir ordinary differential equations. These new equations are 'structurally similar' for all the processes (flow in the soil, over the slopes and in the channel network) and are shown to be representative of the original equations at the new finite pixel scale. With a view to assessing the quality of the approximation, the paper presents a comparison study in which a quasi-analytical approach based upon the characteristic kinematic wave solution of the original equations is compared to the solution provided by the non-linear reservoir finite scale approximation. The test is first applied to a simplified theoretical case in order to show the quality of the approximation at small scales. Successively it is applied to two real-word cases: the Upper Reno catchment based on 400 x 400 m(2) pixels and the Sieve River catchment using 1 x 1 km(2) pixels. All the results show that the cascade of non-linear reservoirs is a good finite dimension approximation of the subsurface flow in the soil, the flow over the slopes and the flow in the channel network, which allows us to properly retain the physical properties of the original equations at finite scales that range from a few metres up to I km