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We develop a maximum-entropy method for streamflow estimation from surface velocities. Entropy maximization, with mean and variance constraints, yields a mass and momentum conserving, error-function-type velocity profile. That sigmoidal profile's infinite branch shapes the hydrometrically critical free-surface region unphysically, so we derive approximate relationships of the ratio of the mean velocity V to the surface velocity v s u r f , f v = V / v s u r f , as function of β = E v 2 / V 2 (E = expectation); β is tightly estimable: β ≈ 1.01–1.15, fv ≈ 0.9–0.72. fv(β) is adaptable to the site geometry and roughness (i.e. to the local hydraulics, instead of the default fv = 0.86), improving the parsimonious discharge estimation from surface velocities. Laboratory data verify the concept. Tests at cross-sections of streams, with known variable bathymetry and roughness, demonstrate the method's ability to estimate the discharge solely from surface-velocity observations, with ± 5% accuracy referenced to discharge determined from densely sampled in-stream velocities. [ABSTRACT FROM AUTHOR]