What the Lyne–Hollick Baseflow Filter Actually Does: Its Frequency Response, and How Much the Baseflow Index of Six US Rivers Depends on One Parameter and the Number of Passes

The baseflow index, the share of streamflow that comes from groundwater and other slow sources, is often computed with the Lyne–Hollick recursive digital filter. I derive the frequency response of the filter's quickflow equation, |H| = ((1+a)/2)|1 - e^{-iw}|/|1 - a e^{-iw}|, which is exactly 1 at a period of two days and 0 at zero frequency. With the usual parameter a = 0.925, the filter passes 99.67% of a 7-day cycle and 93.77% of a 30-day cycle to quickflow, but only 21.57% of an annual cycle; simulations confirm these values. The filter is therefore a high-pass filter whose cut-off sits at periods of a few months, and the split it makes is a choice of time scale, not a physical measurement. Applied to 74 complete water years (1952–2025) of daily flow at six unregulated or lightly regulated USGS gauges, the three-pass baseflow index ranges from 0.370 (St Marys River, Florida) to 0.635 (French Broad River, North Carolina). Changing a from 0.90 to 0.95 lowers the index by 0.058 to 0.143, and using one pass instead of three raises it by 0.143 to 0.278. The ranking of the rivers is stable, and across the six gauges the index has a correlation of 0.855 with the ratio of the 90%-exceedance flow to the median flow, a parameter-free measure of the same property.