Interpreting hydrologic response using transfer function models with time-varying parameters: an example from the Virginia Blue Ridge

Streamflow hydrographs represent the integrated effects of hydrologic processes operating over a wide range of spatial and temporal scales. Constant-parameter transfer function models have been shown to represent the relationship between effective rainfall and streamflow adequately in many cases, but are limited in ability to reveal detailed behaviour by their aggregation of watershed dynamics in a linear, time-invariant model. One way of examining details of catchment dynamics while retaining a simple model structure is to fit linear but time-varying models. We illustrate this approach applied to seven years’ daily rainfall and streamflow data from a 10 km forested watershed in Virginia. Low-order models with an output-offset term (modeled as zero, constant or time-varying) are fitted by extended least squares estimation with optimal smoothing, treating time-varying model parameters as random walks. The extent of time variation is restricted to keep the ratio of mean-squared one-step-prediction error to mean-squared residual close to unity, so that parameter updates track changes in watershed input-output dynamics rather than noise in the record. All models turn out to have one dominant pole, indicating that with the slowest components of the flow record accounted for by an offset, watershed response can be well modeled as a single, linear reservoir with varying gain and time constant. The reservoir time constant varies fairly smoothly between two and thirteen days. Patterns in the evolution of the time constant and steady-state gain correspond to physically interpretable events in the hydrologic record, including snow accumulation and melt and extreme summer storms. Models of the sort presented here have potential applications in baseflow filtering, in revealing subtle changes in hydrologic response, and in identifying anomalies in the records.