Metamodelling: Theory, concepts and application to nitrate leaching modelling

Nutrient fate and balance models for the root zone of agricultural crops are powerful instruments for the assessment of the impacts of agricultural management on nutrient leaching and groundwater quality. Deterministic nutrient balance models allow to estimate the space-time course of nutrients in terms of soil-crop typology, climatic boundary condition and geo-hydrological boundary condition. However, due to computational time and parameterisation constraints (the large number of parameters needed), the use of deterministic modelling in large scale applications or optimisation studies is prohibited. As an alternative, metamodels can be constructed for specific applications from a limited number of deterministic simulations. Presently, no general mathematical definition of a metamodel is available. In this paper, a mathematical definition for a metamodel as well as for its calibration and validation is developed. Next, a metamodelling analysis for the nitrogen leaching model WAVE is presented. In a first step two metamodelling techniques, i.e. multidimensional kriging and neural network modelling were compared on a sub-sample of generated deterministic modelling results. For this application, multidimensional kriging surpasses radial based neural network modelling, in terms of root mean square error, maximal error and model efficiency. In a second step, multidimensional kriging was successfully implemented for the full data set of available deterministic simulations. Metamodels are constructed for the probability of annual nitrate leaching concentration exceeding the 50 mg/l regulatory (legal) end-point. End-point calculations are done taking into account the mineral and organic nitrogen fertilisation amount, the soil and crop typology, and crop rotation typology.