BFS—A non-linear, state-space model for baseflow separation and prediction

Streamflow in rivers can be separated into a relatively steady component, or baseflow, that represents reliably available surface water and more dynamic components of runoff that typically represent a large fraction of total streamflow. A spatially aggregated numerical time-series model was developed to separate the baseflow component of a streamflow time-series using a state-space framework in which baseflow is a non-linear function of upstream storage, an unmeasured state variable. The state-space framework allows forecasting of baseflow for periods with no rainfall or snowmelt and estimation of residence times in contrast to other hydrograph separation models. The use of a non-linear relation between baseflow and storage maintains model performance over a wide range of time scales but will only provide reliable predictions for periods when the rate of streamflow recession as a fraction of streamflow decreases over time.

The baseflow separation model, BFS, is implemented as set of functions in the statistical computing language R. BFS is run using the main function, bf_sep, which reads model input (a time series of streamflow), calculates the baseflow component of streamflow, writes model output to a file, and returns an error to the user to facilitate automated calibration. The function, bf_sep, has six arguments, which a user must enter: a numerical vector with the time series of measured streamflow volume for each time step; a character string, timestep, that has a value of either “daily” or “hourly” indicating the time step; a character string, error_basis, indicating which simulated streamflow components are used for error calculations; a six-element numeric vector, flow, with parameters characterizing streamflow; a six-element vector, basin_char, with parameters characterizing the geometry of stream basin and reservoirs; and a six-element vector, gw_hyd, with hydraulic parameters. The function bf_sep calls a series of other functions to calculate surface and base reservoir storage and fluxes.

Calibration of a non-linear model for baseflow recession must confront three issues. First, baseflow is a component of streamflow, so it is always less than or equal to streamflow but there is no independent standard for the baseflow component of streamflow. Second, optimization routines can converge on a set of model parameters that result in relatively steady but minimal baseflow that does not exceed streamflow, Q, but has a limited dynamic range. Third, the power function used to generate non-linear first-order baseflow recession (dQ/dt)/Q ≠ constant) may only be sensitive to parameters over a limited range of values, which may not be found by optimization routines.

To address these issues, BFS calculates error as the mean of weighted differences between measured streamflow and either simulated baseflow or the sum of simulated baseflow and surface flow as a fraction of measured streamflow. The difference for each time step is weighted by an exponential function of the length of recession for each time step ranging from 0 for periods when streamflow increases and approaching 1 for long recessional periods. The weight is set to 1 for any time step when simulated streamflow exceeds measured streamflow. Error calculation incorporates limited precision of streamflow measurements.

A four-step calibration process was developed to find a set of viable parameters that maximize the baseflow component within the constraints of the conceptual model (a first-order recession rate that decreases during dry periods). BFS was calibrated at 13,208 U.S. Geological Survey streamgages with available daily streamflow records for at least 300 days from water years 1981 to 2020. The total simulated baseflow component as a fraction of streamflow (BFF) was generally less than the baseflow index (BFI) for 8,368 streamgages where BFF and BFI were available. The median difference was BFF–BFI = 0.11. Large differences were most common in the Interior West where streamflow in many rivers is regulated and is generated predominantly by snowmelt. The baseflow separation model generally allocates less streamflow to baseflow than graphical hydrograph separation in snowmelt rivers.

BFS can be used to forecast streamflow during dry periods by using a time series of real-time streamflow with values of Not Available (NA), appended to the time-series to represent missing (future) streamflow values. The forecast skill of BFS was evaluated in terms of difference between simulated baseflow and measured streamflow as a fraction of measured streamflow on the days of the annual maximum recession period at 5,916 of the sites with at least 10 years of record. The median annual error was less than 50 percent at one-half of the sites and generally improved for drier years with longer recession periods.