How Fast Do Rivers Drain? Brutsaert–Nieber Recession Analysis of Four US Rivers, 1951–2025, and How Gauge Noise Flattens the Recession Exponent

When it stops raining, a river is fed by groundwater, and the rate at which its flow falls tells an engineer how long water supply, dilution of effluent and navigation can be sustained. The standard tool is the Brutsaert–Nieber plot, which fits −dQ/dt = aQᵇ to falling limbs of the daily hydrograph. I apply it to 75 water years (1951–2025, 27,394 days each) of USGS daily flow at four unregulated gauges in Maine, Virginia, North Carolina and Texas. The median-regression exponent is 1.18, 1.20, 1.62 and 0.85, and the linear-reservoir time constant from the slowest 5% of recession days is 94.5, 68.9, 102.2 and 116.3 days. I prove two results that bear on how these numbers are read. First, the centred-difference estimate of the decay rate on daily data is exactly 2tanh(1/(2K)), so the discretisation bias in the time constant is about 1/(12K²), below 0.002% here. Second, multiplicative gauge noise flattens the exponent: when the true daily fall is small compared with the noise, the expected observed fall among falling days is 2σQ/√π, proportional to Q, so the fitted exponent is pulled towards 1. In simulation a true exponent of 1.5 is estimated as 1.50, 1.25 and 1.05 with noise of 1%, 2% and 5%. The exponents fitted to the lower half of flows (0.55 to 1.07) are therefore lower bounds on the physics, not evidence of it.