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`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'>..

_{o}Solving for the two wind profile parameters,

`D = z _{1} - (z_{2}-z_{1})/[exp(k(U_{2}-U_{1})/u_{*}) - 1]`

`z _{o} = (z-D)exp(-kU(z)/u_{*})`

`= (z _{2}-z_{1})/[exp(kU_{2}/u_{*}) - exp(kU_{1}/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} = (dz_{2}-dz_{1})/(z_{2}-z_{1}) - (z_{2}-D)/(z_{2}-z_{1})(dU_{2}/U_{2}-du_{*}/u_{*})kU_{2}/u_{*} + (z_{1}-D)/(z_{2}-z_{1})(dU_{1}/U_{1}-du_{*}/u_{*})kU_{1}/u_{*} `

`dD = dz _{1}(z_{2}-D)/(z_{2}-z_{1}) - dz_{2}(z_{1}-D)/(z_{2}-z_{1}) + (z_{2}-D)(z_{1}-D)/(z_{2}-z_{1})*[(dU_{2}/U_{2}-du_{*}/u_{*})kU_{2}/u_{*} - (dU_{1}/U_{1}-du_{*}/u_{*})kU_{1}/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`are proportional to

_{o}`z`, whereas the errors in zero-plane displacement

_{o}`dD`are proportional to

`(z`and

_{1}-D)`(z`). Thus as

_{2}-D`z`, its uncertainty decreases proportionally, whereas the uncertainty in zero-plane displacement does not decrease as

_{o}approaches 0`D approaches 0`. The error terms containing the factor

`kU/u`4-8 can be particularly large. For example, if

_{*}=ln((z-D)/z_{0})~`z`2 and

_{2}/ z_{1 }=`D << z`, then

_{1}`dz _{0} ~ (4-8)z_{0}[dU/U - du_{*}/u_{*}] `

`dD ~ z _{2}[dU/U - du_{*}/u_{*}] `