Optimal Copula Models For The Observed Discharge And Nitrate Concentration In The Lower Illinois River

Investigating the complex relationships between water quality parameters and river discharge has been an area of active research for decades. The objective of this study is to apply bivariate distributions to river nitrate concentration and discharge and to suggest the procedure of choosing the best bivariate distribution in the selected models. Nitrate concentration (NO3) data were measured from 1972 to 2012 at five stations in the Lower Illinois River basin, USA. The bivariate distribution was represented by applying three copula models to explore the dependence between discharge and nitrate concentration. The marginal distributions for the copula models include the generalized Pareto (GPA), generalized extreme value (GEV), two-parameter lognormal (LN2), and three-parameter lognormal (LN3) distributions as well as the three copula models of Clayton, Frank, and Gumbel, respectively. Each procedure was tested by the robust diagnostic, the probability plot correlation coefficient (PPCC) test, and the goodness-of-fit test (statistic). Consequently, the Frank copula model with GPA for stream flow-GPA for nitrate concentration performed more accurately for data from the mainstream stations, Havana (D-31) and Valley City (D-32) and for those from a station on one tributary, Oakford station along the Sangamon River (E-25). However, along other tributaries such as the La Moine River at Ripley (DG-01) and Spoon River at Seville (DJ-08), the Clayton copula model with NL3 for stream flow-GPA for nitrate concentration showed better fit. This study enabled us to develop a procedure to determine an optimal copula for application to the environmental sciences and suggest a case study for the application of copula models to discharge, as a hydrologic variable, and nutrient concentration as a water quality variable. The data files are Excel csv format. The time format is mm/dd/yy. The unit is cubic feet per second (cfs) for flow and mg/L for nitrate-N concentration, stored in five files.