Advances in Quantifying Streamflow Variability Using Hierarchical Bayesian Methods with SPARROW

Open Access
Alexander, Richard Brown
Graduate Program:
Forest Resources
Doctor of Philosophy
Document Type:
Date of Defense:
October 09, 2015
Committee Members:
  • Elizabeth Weeks Boyer, Dissertation Advisor
  • David Russell Dewalle, Committee Member
  • Christopher J Duffy, Committee Member
  • Tyler Wagner, Committee Member
  • hydrology
  • watershed modeling
  • Bayesian statistics
  • streamflow
  • SPARROW modeling
Progress has been made over past decades in synthesizing knowledge of hydrological processes across multiple spatial scales to improve the methods for predicting streamflow in ungauged watersheds. This includes the use of “top-down” modeling approaches that have revealed many of the dominant large-scale controls on water generation and transport. Although informative, further advancements are needed to improve the sensitivity of these models to the range of processes that control streamflow across multiple spatial scales. These include improved methods for quantifying spatial and temporal variability in the parameters and uncertainties of hydrological models, which are needed to enhance the predictive capabilities of hydrological models in ungauged watersheds. To address these needs, hierarchical Bayesian techniques were used with the spatially explicit source-transport model SPARROW (SPAtially Referenced Regressions On Watershed attributes) to predict mean annual streamflow in streams in the continental United States and mean seasonal streamflow in the Susquehanna River Basin (SRB). The models greatly expanded the spatial and temporal range of climate and watershed conditions for assessing the use of hierarchical methods in hydrological modeling. Hierarchical techniques were used to estimate streamflows according to a multi-level model in which a multivariate model of the parameters was nested within a model of the streamflow observations. Using a separate parameter model had the advantage of allowing model coefficients and the model error variance to vary regionally or seasonally, thereby increasing the sensitivity of the model to process controls that operate over these spatial and temporal scales. This was enabled by Bayesian simulation of the hierarchical parameters, which allowed the parameters to be described as random variables with a probabilistic joint distribution of likely values that may vary over space and time, rather than being constrained to fixed but unknown quantities as is typical of non-Bayesian optimization methods. Both the hierarchical and Bayesian features promote the notion of data sharing or information “borrowing”, in which the Bayesian model parameters represent a weighted combination of the parameters from two extreme conceptual modeling approaches: one reflects the use of unique watershed-specific models, whereas the other reflects the use of a single universal multi-watershed model. The former conceptual model is sensitive to unique local hydrology, whereas the latter describes global hydrological relations that are shared across all watersheds. Statistical advantages arise from integrating these two approaches in cases where the local models suffer from less precise, data-poor information, given that the global model is likely to be informed by more statistically precise (but perhaps locally biased), data-rich information for a large collection of watersheds. In addition, a more complex hierarchical state-space Bayesian model was used to unravel the sources of error that are responsible for prediction uncertainties, based on a simultaneous accounting of uncertainties in streamflow observations, parameters, and structural components of the model. Accordingly, a model equation was specified for the streamflow observations and their uncertainties (i.e., “measurement errors”), within which a second equation was nested that described the deterministic structure of the model, including the river network, explanatory variables and associated coefficients, and a process-related variability component (“process uncertainties”). The process uncertainties quantify process-related variations in streamflow that are unexplained by the measurement errors and the structural representation of physical processes in the model. The process uncertainties may include random and systematic variations in hydrological processes, such as space/time biases, heteroscedasticity, nonstationarities, and spatial or temporal correlation. The measurement errors are expected to primarily reflect model uncertainties primarily associated with the measurement of streamflow (e.g., water velocity, rating curve stability). However, measurement errors are potentially sensitive to inaccuracies in locating monitoring sites on reach segments or localized influences on flow (e.g., diversions, gaining/losing streams) that are not recognized by the deterministic model structure and not accounted for in the downstream observations of flow within the hydrologically nested array of monitoring sites. Separation of the sources of uncertainties was facilitated by the spatial propagation of the process uncertainties (and exclusion of the measurement errors) during the model accumulation of water volumes from individual catchments within the river network. The process uncertainties and measurement errors are specified so that they vary over space and time and quantify uncertainties associated with the smallest observational units in the model—i.e., seasons and incremental drainage areas between monitoring sites (interquartile range of 475 to 3,225 km2 for the national model and 183 to 1,075 km2 for the SRB model). The state-space model led to improved model accuracy and interpretability and revealed information about the potential causes of model uncertainties. Predictions of streamflow, updated to include the state-space estimates of the process uncertainties, were designed to provide an improved measure of the latent “true” streamflows, which was tested in model comparisons that were conducted for a set of validation monitoring sites. The application of hierarchical Bayesian methods to the SPARROW model in this study addressed three questions: (1) What are the natural and cultural factors that control spatial and temporal variability in mean annual streamflow in watersheds of the United States and mean seasonal streamflow in the SRB? (2) Is there evidence for hierarchical (multilevel) effects in the SPARROW streamflow models over large spatial scales? (3) What are the effects on model accuracy and process understanding of using a more precise characterization of the streamflow model uncertainties, based on hierarchical Bayesian state-space methods? In relation to the first question, the SPARROW model was sequentially coupled with a conceptual monthly water-balance model of the “natural” effects of water supply and demand on unit-area runoff. Mean-annual and mean-seasonal streamflow were estimated a function of aggregate-monthly inputs of unit-runoff, mediated by additional natural and anthropogenic properties that control water availability, demand, and transport. The coupled models markedly improved prediction accuracy and identified additional climatic, terrestrial, and aquatic (reservoir, stream channel) controls on spatial and temporal variability in streamflow. One unique finding is that water losses in streams, expressed as a fraction of the water volume per day of water travel time, were estimated to be larger in small streams and in western regions; the losses may stem from a combination of processes, including direct evaporation from water surfaces, recharge to the subsurface (especially in western streams), consumptive loss from diversions, and transpiration from riparian vegetation. In relation to the second question, accounting for spatial and temporal variability in model coefficients and model error variance improved prediction accuracy and enhanced understanding of the scale-dependence and potential causes of model uncertainties. The national model was found to display generally modest regional variability in coefficients; however, accounting for regional and drainage size-related variations in model uncertainties, which were positively correlated with the extent of aridity in the watersheds, resulted in large improvements in prediction accuracy in eastern watersheds. In the SRB model, accounting for intra- and inter-annual variability in the unit-runoff coefficients led to larger improvements in prediction accuracy than the addition of spatial variability in other model coefficients. This more temporally complex model led to larger improvements in accuracy in streamflow predictions during the more extreme water supply and demand conditions in the spring and summer seasons and especially during wet and dry years. As to the last question, the state-space methods led primarily to improved characterizations of the process-related model and prediction uncertainties, with generally more modest effects observed on the values of the model coefficients and their levels of precision. Several key patterns emerged in the process uncertainties and measurement errors. First, process uncertainties were estimated to be larger than the measurement errors, owing to the relatively small sampling and analytical errors that were expected to be associated with the streamflow observations. Second, the measurement errors and process uncertainties displayed similar regional patterns in the national model, with larger values in the more arid western regions. Third, the larger process uncertainties in the state-space models suggest that the uncertainties associated with the more conventional (non-state-space) SPARROW streamflow model were generally underestimated. Finally, evaluations at validation sites in the SRB confirmed the markedly improved accuracy of the state-space seasonal streamflow predictions; these predictions were updated to include the estimates of process uncertainties, which displayed a strong spatial covariance structure. The gains in accuracy were larger during summer seasons when the dominant effects of groundwater inflows and transpiration losses are likely to cause a stronger spatial correlation structure in the process uncertainties. Gains in accuracy were smaller during spring seasons when a diversity of hydrological processes can occur, including snow melt events and periods when soil and sub-surface water storage is reduced by evapotranspiration. By contrast, predictions from the national state-space model at validation sites showed only limited gains in accuracy in a few western regions, suggesting that the process uncertainties in mean annual streamflows are spatially heterogeneous and lack a detectable spatial covariance structure in most regions. The larger size of these drainages in the national model may have contributed to this pattern. This stresses the possible importance of using more spatially resolved river networks (as used in the SRB) and larger numbers of historical daily monitoring stations in future hierarchical modeling studies. The results of the study advance hydrological understanding by providing important insights about major process controls on streamflow variability in predominantly developed and climatically diverse watersheds. The evaluations of the hierarchical Bayesian methods demonstrated their effectiveness for enhancing model sensitivity to hydrological processes that operate over multiple spatial and temporal scales. This included the use of spatially and temporally varying model parameters that identified regional and seasonal/annual uniformities in hydrological responses to major controlling factors, including climate, land use, and stream channel processes. The evaluations also identified methods for unraveling the sources of model uncertainties that revealed process-related information on the seasonal spatial-correlation structure in streamflows; explicitly accounting for this structure markedly improved prediction accuracy at monitoring sites used to validate the model. These techniques can be used to enhance the performance of other statistical and mechanistic hydrological models. They can also advance methods for quantifying hydrological similarities among neighboring watersheds, which are needed to reliably transfer hydrological data and models to ungauged catchments (e.g., classification techniques). The continued development of the model specifications from this study has the potential to contribute to further advances. These potentially include the simultaneous modeling of surface and base flow in streams and the addition of functional specifications of the process uncertainties to explicitly account for aridity-related spatial variability and spatial and temporal covariances in streamflows.