Abstract. The fully rough form of the law of the wall is commonly used to quantify velocity profiles and associated bed shear stresses in fluvial, aeolian, and coastal environments. A key parameter in this law is the roughness length, z₀. Here we propose a predictive formula for z₀ that uses the amplitude and slope of each wavelength of microtopography within a discrete-Fourier-transform-based approach. Computational fluid dynamics (CFD) modeling is used to quantify the effective z₀ value of sinusoidal microtopography as a function of the amplitude and slope. The effective z₀ value of landscapes with multi-scale roughness is then given by the sum of contributions from each Fourier mode of the microtopography. Predictions of the equation are tested against z₀ values measured in  ∼ 10⁵ wind-velocity profiles from southwestern US playa surfaces. Our equation is capable of predicting z₀ values to 50 % accuracy, on average, across a 4 order of magnitude range. We also use our results to provide an alternative formula that, while somewhat less accurate than the one obtained from a full multi-scale analysis, has an advantage of being simpler and easier to apply.