Quantifying uncertainty in flowrate modelling using spatially defined fuzzy entropy based on hydrological processes in a catchment

• Fuzzy entropy quantifies spatial data uncertainty in a watershed. • A new fuzzy entropy calculation method based on hillslope flow processes is proposed. • A new normalized fuzzy entropy parameters indicates the appropriate watershed scale for modelling runoff. • Complements Monte Carlo analysis with higher computational efficiency and upscaling insight. A method is proposed that uses hillslope hydrological processes to develop measures of fuzzy entropy distribution over a landscape, in order to estimate the uncertainty arising from the spatial distribution of data input to runoff models. How this distribution impacts the upscaling process in watershed hydrological simulations is also explored. Spatially distributed membership functions based on the distribution of numerical (slope) or categorical (landuse and soil type) spatial data inputs to a hydrological model are derived. Fuzzy inferencing that incorporates expert knowledge of flow mechanisms in a watershed is used to create a new variable referred to as runoff potential. Spatial distributions of fuzzy Shannon entropy S F ( μ ij ) are developed and two new parameters: the watershed fuzzy Shannon entropy W S F ( μ ij ) and the normalized watershed fuzzy Shannon entropy W S F ^ μ ij , are proposed to quantify the uncertainty in runoff potential given the spatial distribution of the input data as it changes through the catchment along flowpaths. Comparing the proposed fuzzy entropy-based method with a traditional Monte Carlo method applied with PCSWMM model simulations demonstrates that the proposed method provides additional insights into how uncertainty is generated spatially that a conventional approach cannot provide; while significantly improving computational efficiency. In the application to a 17.46 km 2 , mixed-landuse catchment, W S F ^ μ ij fluctuated greatly at small scales but then reached a stable, constant value within approximately 18.5 % of the total catchment area. Thus, revealing how uncertainty in spatially-scaled processes propagates along hydrological pathways, which thereby provide a reference for model optimization and water resources management at the basin scale.