A Bayesian hierarchical approach to combine interpolation and extrapolation methods for fluvial flux calculation

• New method combines extrapolation and interpolation methods to improve fluvial flux calculations. • Method uses a Bayesian hierarchical approach to combine categorical and continuous variables. • The method is more precise than previous methods without requiring any more data. • National-scale results are presented for suspended solids and nitrate fluxes from England. Calculation of the flux of material through catchments of all scales is a key measure used to understand how our environment functions and how it is changing. Methods of fluvial flux calculation have tended to be either based upon extrapolation against a well-known driving variable, typically river discharge; or interpolation where expected values (e.g., arithmetic mean) are used. The choice of method has been a matter of personal choice and perception of strength of controls on the determinand of interest. In this study we propose that this dichotomy of methods is unnecessary and show how a Bayesian hierarchical method can combine approaches. Our method uses factors of month, year and monitoring site to modify the concentration-discharge (C-Q) relationship for the determinand of interest, i.e. the method uses all the available data to define the C-Q relationship at each site for each level of the included factors. If the C-Q relationship is not significant, then the method returns the expected value for the particular month, year and monitoring site combination. The method was applied to 161 catchments across England for both suspended solids (SS) and nitrate (N-NO 3 ) concentrations, and results compared to existing extrapolation methods. The method developed here demonstrates: i) Greater sensitivity in that it was better able to identify significant C-Q relationships. ii) The ability to detect smaller significant slopes for C-Q relationships and with improved precision by at least 76%. iii) Scaling results to the national level implies that existing extrapolation methods would overestimate national fluxes. The proposed method uses no more data than would be used by either an extrapolation method or an interpolation method but draws upon the information from all data to strengthen the analysis.