Using Bayesian hierarchical models to better understand nitrate sources and sinks in agricultural watersheds

Abstract

Export coefficient models (ECMs) are often used to predict nutrient sources and sinks in watersheds because ECMs can flexibly incorporate processes and have minimal data requirements. However, ECMs do not quantify uncertainties in model structure, parameters, or predictions; nor do they account for spatial and temporal variability in land characteristics, weather, and management practices. We applied Bayesian hierarchical methods to address these problems in ECMs used to predict nitrate concentration in streams. We compared four model formulations, a basic ECM and three models with additional terms to represent competing hypotheses about the sources of error in ECMs and about spatial and temporal variability of coefficients: an ADditive Error Model (ADEM), a SpatioTemporal Parameter Model (STPM), and a Dynamic Parameter Model (DPM). The DPM incorporates a first-order random walk to represent spatial correlation among parameters and a dynamic linear model to accommodate temporal correlation. We tested the modeling approach in a proof of concept using watershed characteristics and nitrate export measurements from watersheds in the Coastal Plain physiographic province of the Chesapeake Bay drainage. Among the four models, the DPM was the best--it had the lowest mean error, explained the most variability (R = 0.99), had the narrowest prediction intervals, and provided the most effective tradeoff between fit complexity (its deviance information criterion, DIC, was 45.6 units lower than any other model, indicating overwhelming support for the DPM). The superiority of the DPM supports its underlying hypothesis that the main source of error in ECMs is their failure to account for parameter variability rather than structural error. Analysis of the fitted DPM coefficients for cropland export and instream retention revealed some of the factors controlling nitrate concentration: cropland nitrate exports were positively related to stream flow and watershed average slope, while instream nitrate retention was positively correlated with nitrate concentration. By quantifying spatial and temporal variability in sources and sinks, the DPM provides new information to better target management actions to the most effective times and places. Given the wide use of ECMs as research and management tools, our approach can be broadly applied in other watersheds and to other materials.

Leveraging Spatial and Temporal Variability to Probabilistically Characterize Nutrient Sources and Export Rates in a Developing Watershed

“…While these approaches indicated the importance of hydroclimatology for driving interannual variability, they did not explicitly model the effects of hydroclimatological variables on source export rates or other watershed processes. Xia et al () considered land use change and four different statistical approaches for representing the interannual variability in watershed nitrate loading within a hybrid watershed model but did not model the effects of hydroclimatological variables on source export rates. In general, both of these previous studies suggested that dynamic parameter estimation was effective for simulating temporal variability in instream nutrient loads.…”

“…Dynamic parameter estimation allows model coefficients to vary over time in order to increase overall model fit (Congdon, ). In these studies, dynamic coefficients are compared to hydrometeorological variables through post hoc analyses (Wellen et al, ; Xia et al, ), rather than being directly integrated into the deterministic form of the model, as in our study (equation ). While both approaches have merit, directly integrating changes in both precipitation and source distributions into the hybrid model allows us to assess how these factors interact to influence nutrient loading and provides a framework for probabilistically evaluating different climate and land use scenarios.…”

“…Additional factors, beyond land use change and precipitation-induced nutrient wash-off, may also affect the interannual variability of instream nutrient loads. The previous studies by Wellen et al (2012) and Xia et al (2016) suggest that changes in instream residence time and nutrient retention may also play an important role. In this study, we estimated retention based on available NHD stream characteristics, which reflect average conditions over time.…”

“…Hybrid models, on the other hand, can provide an estimate of mean loading rates over large regions through empirical estimation based on available water quality data, watershed characteristics, and land use data (Alexander et al, ). Statistical approaches can also quantify uncertainty in EC estimates, which is important in assessing the reliability of model estimates for considering risk in watershed management (Reckhow, ; Xia et al, ).…”

“…Addressing temporal variability in loading estimates can facilitate understanding of the causes of adverse loading conditions, which can inform more effective management strategies (McClain et al, ). Wellen et al () added temporally variable climatic variables as an additive component to the SPARROW regression, and Xia et al () included land use change. In addition, both of these studies used dynamic parameter estimation, allowing relationships to vary over time.…”