Improving estimates of phosphorus loads using hierarchical models

Abstract

Estimates of nutrient loads are critical for managing the effects of nutrient pollution in receiving waters. An accurate calculation of load requires sub-daily, concurrent measurements of flow and nutrient concentrations. Continuous flow measurements are common, but nutrient concentrations are often measured infrequently, and so, statistical estimates of flow – concentration relationships are often used to infer nutrient concentration on unsampled days. Contributions of diffuse, nonpoint sources to flow – concentration relationships in neighboring sites may be similar, and methods are available to apportion in-stream concentrations to contributions from point and nonpoint sources. Combining a source apportionment model with a hierarchical statistical model may be a useful approach for leveraging information from more intensively sampled sites to improve estimates of relationships at more sparsely sampled sites. The approach is tested using daily measurements of total phosphorus and flow from eight rivers in Ohio, USA. Data from seven of the rivers are randomly subsampled to a single sample from each month, and a hierarchical model fit to estimate flow – concentration relationships for all sites. Hierarchical model estimates of phosphorus loads in subsampled sites were compared with three site-specific models: a linear flow – concentration relationship, a quadratic flow – concentration relationship with seasonal corrections, and Beale’s ratio method. The hierarchical model improved substantially on the precision of estimated total phosphorus loads at sparsely sampled sites, yielding an interquartile range of prediction errors of 22%. Simple linear load – concentration models were the next best performing method with an interquartile range of 33%. The average bias in hierarchical model predictions of phosphorus loads was 8%, whereas the smallest biases associated with the comparison methods were –10% for both the linear load – concentration model and Beale’s ratio. These results suggest that a hierarchical statistical model can be valuable for improving nutrient load estimates when data from a small number of intensively sites are available.