The aerodynamic surface roughness length, z_o, and zero-plane displacement, D, are defined for a neutral wind profile by

U(z) = (u_*/k)ln((z-D)/z_o)

where U is the mean wind speed measured at height z (above ground level), u_* is friction velocity, and k=0.4 is the von Karman constant. D and z_o are best calculated with a wind profile measured at several levels, but (theoretically) they can also be calculated from two levels of wind data as commonly measured by the ISFF Flux-PAM stations, i.e. with a prop-vane at 10m height and a sonic anemometer at a second height.  A good test for neutral stability is zero heat flux <w'tc'> or zero temperature variance <tc'tc'>..

Solving for the two wind profile parameters,

D = z₁ - (z₂-z₁)/[exp(k(U₂-U₁)/u_*) - 1]

z_o = (z-D)exp(-kU(z)/u_*)

= (z₂-z₁)/[exp(kU₂/u_*) - exp(kU₁/u_*)]

Note that if the zero-plane displacement height is known, then the roughness length can be calculated from measurements of wind speed and friction velocity by a singe sonic anemometer.

The sensitivity of these formulas to errors in height, wind speed, and friction velocity measurements are,

dz_o/z_o = (dz₂-dz₁)/(z₂-z₁) - (z₂-D)/(z₂-z₁)(dU₂/U₂-du_*/u_*)kU₂/u_* + (z₁-D)/(z₂-z₁)(dU₁/U₁-du_*/u_*)kU₁/u_*

dD = dz₁(z₂-D)/(z₂-z₁) - dz₂(z₁-D)/(z₂-z₁) + (z₂-D)(z₁-D)/(z₂-z₁)*[(dU₂/U₂-du_*/u_*)kU₂/u_* - (dU₁/U₁-du_*/u_*)kU₁/u_*]

where dz, dU, and du_* are measurement errors in height, wind speed, and friction velocity. It can be noted that the errors in roughness length dz_o are proportional to z_o, whereas the errors in zero-plane displacement dD are proportional to (z₁-D) and (z₂-D). Thus as z_o approaches 0, its uncertainty decreases proportionally, whereas the uncertainty in zero-plane displacement does not decrease as D approaches 0. The error terms containing the factor kU/u_*=ln((z-D)/z₀)~ 4-8 can be particularly large. For example, if z₂ / z₁ = 2 and D << z₁, then

dz₀ ~ (4-8)z₀[dU/U - du_*/u_*]

dD ~ z₂[dU/U - du_*/u_*]