Bayesian modeling of the assimilative capacity component of nutrient total maximum daily loads

Implementing stream restoration techniques and best management practices to reduce nonpoint source nutrients implies enhancement of the assimilative capacity for the stream system. In this paper, a Bayesian method for evaluating this component of a total maximum daily load (TMDL) load capacity is developed and applied. The joint distribution of nutrient retention metrics from a literature review of 495 measurements was used for Monte Carlo sampling with a process transfer function for nutrient attenuation. Using the resulting histograms of nutrient retention, reference prior distributions were developed for sites in which some of the metrics contributing to the transfer function were measured. Contributing metrics for the prior include stream discharge, cross‐sectional area, fraction of storage volume to free stream volume, denitrification rate constant, storage zone mass transfer rate, dispersion coefficient, and others. Confidence of compliance (CC) that any given level of nutrient retention has been achieved is also determined using this approach. The shape of the CC curve is dependent on the metrics measured and serves in part as a measure of the information provided by the metrics to predict nutrient retention. It is also a direct measurement, with a margin of safety, of the fraction of export load that can be reduced through changing retention metrics. For an impaired stream in western Oklahoma, a combination of prior information and measurement of nutrient attenuation was used to illustrate the proposed approach. This method may be considered for TMDL implementation.